Einstein and "spooky actions"

[Mr Spinkles]

A few responses to some things:

First: if you don't think measurements are represented by operators acting on kets, you are simply wrong. It's stated unambiguously in Shankar for example

You mean on page 124 ...

I mean on p. 116, under Postulates: "The independent variables x and p of classical mechanics are represented by Hermitian operators ... the operators corresponding to dependent variables w(x,p) are [also] given Hermitian operators Omega ... measurement of the variable (corresponding to) [the Hermitian operator] Omega will yield one of the eigenvalues w with probability ... (etc., etc.) ... the state of the system will change from |Psi> to |w> as a result of the measurement". And p. 122 "the measurement of the variable Omega [Hermitian operator] changes the state vector ... this phenomenon is called the collapse or reduction of the state vector".

As long as we are quoting Shankar, let's compare the following statements:

In classical mechanics the mathematical representation is a list of all the positions and momenta of all the particles, i.e. a point in a 6N-dimensional phase space; in QM it's a vector in Hilbert space.

The Postulates:

Classical Mechanics I: The state of a particle at any given time is represented by the two variables x(t) and p(t), i.e., as a point in a two-dimensional phase space.

Quantum Mechanics I: The state of the particle is represented by a vector |Psi> in a Hilbert space.

I see no meaningful difference between what I said, and what Shankar says. (Note that he refers to a point in a 2-dimensional instead of 6N-dimensional phase space because he is only considering one particle in one dimension.) Do you really maintain that this is not "a pretty unassailable statement" and that this would be a controversial thing to say in a room full of physicists?

As Heisenberg's uncertainty principle tells us, we cannot ever have complete knowledge of a quantum physical system (which you are well are of). So how can we mathematically represent our complete knowledge of the state of a system that we claim it is impossible to know the exact state of?

I think you are confused by the term "complete knowledge". If I don't know what color an insect's seventh leg is, or what is the temperature north of the North pole, is my knowledge necessarily incomplete? Or could I have complete knowledge about a universe which does not have insects with seven legs and places north of the North pole? Even in classical mechanics, with ordinary classical waves, there is an uncertainty principle, because mathematically and physically, a wave cannot have a perfectly well-defined position and momentum simultaneously. This does not prevent us from knowing everything there is to know about the wave, which is what I would have meant by "complete knowledge".

Furthermore, uncertainty principles (classical and quantum) do not prevent us from knowing "the exact state" of the system (emphasis added). In wave mechanics, the state is described by, for example, y = sin(x). In QM, the state is described by a wavefunction or ket. We can know the exact state, even if this state does not possess a precise position and momentum simultaneously, in accordance with an uncertainty principle..

So, let's not worry about the semantics of "complete knowledge". Let's just distinguish between two cases: (1) a particle can have a definite position and momentum simultaneously, but I only know about one or the other; (2) a quantum particle, like a classical wave, cannot have a definite position and momentum simultaneously; hence, I can know everything there is to know, but I can never obtain a definite position and momentum simultaneously.

For starters, before we continue, do you acknowledge the distinction between these two cases?

Allow me to clarify: they don't correspond to measurements in the classical sense.

Of course! Clarification accepted.

I simply used the term as it has been even in statistical mechanics. A physical system is deterministic in principle. I have no qualms with asserting that physical systems can be really, truly indeterministic, and that ...

I find this confusing. The second sentence is simply not true, by definition. The third sentence seems to contradict the second. Is it safe to disregard the second sentence and accept our mutual agreement that physical systems could (in principle) be indeterministic?

But what do we have complete knowledge of? It might be the physical system, as you posited, or not. But until you can demonstrate any evidence anywhere that QM posits complete knowledge of a physical system as the standard interpretation and that this physical system has a one-to-one correspondence with the mathematical representations, then statements like this ... give me nothing.

Are you suggesting that the ket represents the complete state of an unphysical system? (The part about the ket is my claim, BTW, and my preferred way of saying it). One would hope that it goes without saying, in a theory of physics described in a physics textbook, that the thing we are trying to represent is physical--an electron, for example, or an atom, etc. By assumption.

 

[LegionOnomaMoi]

I'm glad buying a used textbook had something come out of it, and particularly glad that Shankar has that side-by-side comparison.

I mean on p. 116, under Postulates:

"The independent variables x and p of classical mechanics are represented by Hermitian operators X and P with the following matrix elements in the eigenbasis of X

the operators corresponding to dependent variables

are given Hermitian operators

That's the 2nd postulate for QM. For classical, we have only any given dynamical variable is a function of time and space. It's extremely intuitive and quite easy: some system changes over time as a function of velocity and position. This is simple not just because of this postulate itself, but because the first classical postulate gives us a general description of initial conditions. The system is described dynamically by using time as the argument for the position and velocity functions. However, we have no such conditions in QM. We have a vector in Hilbert space.

Also note that the differences in description. For classical (emphasis added):
"the state of the particle at any given time is specified..."
vs.
"The state of the particle is represented by..." So we have a dynamical system In both cases, yet no initial conditions in the first. We have a mathematical representation before we have any actual knowledge of the system. In the second postulate we represent the classical variables with Hermitian operators. We have a system in Hilbert space and mathematical operators give us the classical variables. We can see how this plays out in measurement with the 3rd postulate:

"If the particle is in a state |Ψ>, measurement of the variable (corresponding to) Ω will yield one of the eigenvalues ω with the probability

The state of the system will change from |Ψ> to |ω> as a result of measurement."


So in the QM case, we represent the system by a vector, while in the classical case we are given position and velocity as functions of time. That makes the next step for the classical case easy. Start at time t nought, measure the two variables at this time, and then let it run. It's a dynamical system, as "the state variables change in time according to Hamilton's equations", so we plug the state variables into these.

In the QM case, we can't do that. We have an entire additional level of formalism in order to get to measurement, because we begin with a mathematical representation of the system that is determined by theory, experimental design, and preparation, but is never checked against any system other than itself (i.e., there is no known direct relationship between the representation go the system and the system). To obtain the equivalent state variables in the QM case, we rely on mathematical operators constructed (like the state of the system) from theory and design, but independent of measurement. We know this because Shankar states "If the particle is in state |Ψ>" as the opening of the 3rd postulate. That's a ket. It is supposed to be a mathematical representation of our compete knowledge of the system. So why are we asking whether the particle is in a particular state after measurement?


Notice too that here the dynamical variable is our mathematical operators that allow us to relate what they correspond to or represent to probabilities. The "observables", represented by the Hermitian operators, don't correspond to an observation but the probability of one.

I see no meaningful difference between what I said, and what Shankar says. (Note that he refers to a point in a 2-dimensional instead of 6N-dimensional phase space because he is only considering one particle in one dimension.)

Note that you describe the two (classical & quantum) mathematical representations as basically equivalent: mathematical representations of some system. Shankar doesn't. For classical, he says we are given the state at any specified time. For QM, we represent it. At no place in the postulates of classical mechanics do we "represent" anything in Shankar's schema. But he continues to use "represent" for QM. Why? More importantly, recall this:

Physical reality: some physical system (e.g. an electron)
Mathematical representation: ket

Physical reality: physical measurement of a system
Mathematical representation: operator "hitting" a ket

Where, in the schema Shankar uses, does he state that the operator is the mathematical representation of measurement? He doesn't. In fact, we have them before measurement. Even better, our ket (our complete knowledge) requires us not only to use such an operator, but it is only if the particle is in the state |Ψ> that we can use the operator to tell us the probability of measurement.

Do you really maintain that this is not "a pretty unassailable statement" and that this would be a controversial thing to say in a room full of physicists?

I think that what Shankar states is the standard and most accepted one, but whether it would be controversial depends on whether a single person yells out "is the moon there when you don't look at it?". I agree completely that the schema Shankar lays out is the most accepted one and regardless of ontological questions, it has been extremely successful.

I don't think that he says what you did. I think the use of represent for QM, the use of "represents" is deliberate, I think that the use of operators is not as you described (they are not the mathematical representation of measurements but an independently developed formalism used to relate the specifications of a prepared quantum system to outcomes probabilistically. I do not see the one-to-one correspondence between the mathematical representation and the system that I said is unknown and you stated was not.

I think you are confused by the term "complete knowledge".

I don't actually like that term. I would say "contains all the information about the system". The postulate is that the wave function or ket contains all the information there is about the system. What this actually means, however, continues to be much debated.

Alas, I have to go to an after funeral gathering and cannot respond to the rest. However, this may be useful as I seem to have implied what I did not intend by complete knowledge. I have read "the information is contained in", "all the information there is about the system is contained in", and other variations. I don't know if that clarifies at all, and I apologize for not finishing.

Also, I can't thank you enough for bearing with me. I really wish I had your composure.

 

[PolyHedral]

"There are three basic kinds of operations that can be used to solve any system of linear equations. These operations are simple transformations of the original system of equations that preserve the information content about the solution. "
[..]
Note that contrary to whatever you think you were claiming, the transformations resulting from row operations preserve the original matrix.

You're doing it again! You're quoting things that look like they back you up, but actually don't.

Row interchange preserves the system, but not the matrix. I proved that earlier - equal matrices subtract to form the zero matrix, and those ones - which do represent the same system - did not.

Otherwise, why the hell would we use them to solve systems? Matrices aren't filled with scalars but coefficients. How do you add or subtract the coefficients without any variables? You don't.

If we're interpreting the matrix as a system of equations, the coefficients are still there - I just adjusted the two equations in the first matrix using operations I know are arithmetically valid.

There is no "the set V"

The definition of an n-dimensional vector space over the field F is the set,

This means that every single vector space other than the empty space is defined by some collection of vectors whose linear combinations span that space. Linear combinations include both vector addition and scalar multiplication. Ergo, YOU CANNOT HAVE a vector space V without scalars!

...with the operations,

Notice how lambda is not a member of V, and yet there you have your linear combinations: from the statement above, it's almost trivially obvious that:

(where e_i = [0, 0, ..., 0] except in the ith place, where it is 1)

I have a vector space built on top of a field of scalars, but in almost all cases, the scalars are not part of the vector space. They are not a member of the set V, except in the trivial case when n=1. (Where there is a bijective map between the two.)

I need only two vectors two span R2. The vector space, however, is not these two vectors. It is their linear combination, and thus the elements that make up the space include infinitely many scalars.

So which element of

is also a member of R?

You will never understand the concept of vector spaces unless you have a firm grasp of what determines the dimensionality of a space and a vector because you will keep talking about vectors existing in spaces they can't.

I can define strongly-typed, fully general vector spaces a priori and deduce from those axioms. You, OTOH, are quoting textbooks that don't agree with you and making assertions with obvious counterexamples. Where are your proofs?

Intuitively" is not formally. Why might this be important?

Because were it true, standard formulations of Boolean algebras or any algebraic structure which used set theoretic operators: union and intersection for addition and multiplication, respectively

.
You did see the quote earlier that directly contradicted the idea that Boolean algebras are field structures, didn't you?

Boolean algebras, if you recall, are fields. They are created by taking an arbitrary set S, defining union, intersections, and the unary operation complement, and the power set algebra on S

Can you prove that this is a field, i.e. it has satisfactory addition and multiplication operators?

Anyway, you don't need multiples of 2 but you do need to have n greater than or equal to 2. The trick then is to take n finite integers creating a finite field by using the value of n.

The set formed by the values of [0,n-1] is a very different one than that defined by the integral multiples of some n. The former is a field for prime n, but I don't think the latter is a field for any n != 1.

What V? And do you mean a field over a vector space (in which case, once again, the answer is yes) or are you asking if a vector space is a field? That second question relates to this:

The question relates to whether V is a field. Asking the question is equivalent to asking whether V as a set has operators that fufill certain predicates.

It's equally meaningless, if that's what you mean. The above says that the addition of two functions that maps nothing to a variable x which maps nothing from nowhere to an unspecified set.

It maps two functions to a single function, which can be inferred to return a value of the same type as the two functions when given a value within the intersection of the two functions' domains.

This is how the second arrow is used. It defines a general function not in terms of values but in terms of spaces or sets

Did you notice that we used two different arrows? Also, regardless, are you aware of what "operator overloading" refers to?

Think of the Cartesian plane as R2. What defines a vector here? an increment, or directed amount of change, that is associated with a point R x R. Every vector in R2 is associated to a point (x, y). If you have a vector with coordinate entries x, y, & x, then where in R2 can they go?

If you define a plane in R3, then there will be a bijective map (i.e. a homeomorphism) from that plane to R2. That's what I mean when I say that 2 n-D vectors will span R2.

 

[LegionOnomaMoi]

Row interchange preserves the system, but not the matrix.

Now, swap the rows:

This is clearly the same system of equations - after all, the answers we get don't depend on the other we write the equations

They are not identical..

1) Look up "augmented matrix" and "coefficient matric" for a refresher
2) The matrix is a notational variant of the system. What you've done above is left out the bar to indicate an augmented matrix, violated the logic by applying valid row operations on one system and then acted as if the resulting matrices were now two independent entities you can perform operations on.
3)
Let
5x - 3x = 2x
define a valid equation. Then
-3x +5x = 2x
&
-6x + 10x = 4x

are both ways to re-write the original equation while ensuring they are equal. Unless we use your logic, which is to ignore what the notation is there for and adopt personal definitions instead.

Matrix notation has a logic of its own that e.g., allows certain row operations s.t. the underlying algebraic structure is preserved. You not only didn't use the correct notation to begin with but followed that up by treating the row swap as creating a new matrix s.t. you could apply matrix subtraction of a system to itself. You didn't start with the correct notation, didn't apply valid operations (row swapping is done on the same matrix; it doesn't make a new one you can subtract from the first) and as a result your conclusions are false.

Row operations on a matrix do not create new matrices. Perhaps had you looked into how a coefficient matrix and an augmented matrix differs from the matrices in e.g., equations such as Ax=b, you wouldn't have made this error, which is essentially this:
Let
5x - 3x = 2x
then
5x - 3x 2x -3x+5 2x
it's meaningless.



You get your inequality by applying a personal definition to a standardized notation.

The definition of an n-dimensional vector space over the field F is the set,

A vector in n-dimensional space requires n entries for every vector. R3 has 4 types of spaces (and every vector in R3 is a column vector with 3 rows): R3 itself (with basis vectors s.t. the equation Ax=0 has only the trivial solution), infinitely many planes in R3, infinitely many lines in R3, & 0.

Vector field extension is the application of operators F to vector spaces by defining binary operations: multiplication by scalars & addition of vectors. By doing this, we have created an algebraic structure for some V. Before you can do this, you should show that you actually have a vector space at all.

with the operations,

The top part could actually be meaningful without the novel field/vector notation. However, you end up with a single vector that may or may not be a basis vector and which exists only in n-dimensional space. The bottom is meaningless. You just put a bunch of symbols together that are similar to a real mathematical system.

from the statement above, it's almost trivially obvious that

..it's wrong. Every e in that equation is a vector with n entries (all having a single nonzero entry of 1). So if my matrix is an n by n matrix, I can multiply it by a n-dimensional standard basis vector e1 and end up with the first column of my matrix. If I multiply an n by n matrix by a n basis vectors all of n dimension then I will end up with my original matrix.

They are not a member of the set V

What are the members of "the set V"?

I can define

I can define addition as always pairing four arbitrary symbols and returning III, but it is pointless. You are not working within any system you've shown exists, whether an algebraic one or not. Your equation above says an n dimensional vector is equal to an n by n matrix. Had you kept the right side as is, but on the left side multiplied the n-dim vector by the n by n identity matrix the equation would've been true.

You, OTOH, are quoting textbooks

On the topic of QM, yes. Here, I've gone out of my way to provide you with sources. You have cited any such that support your definitions or conclusions. I program using an environment that is so built around matrices its actually called matrix lab (MATLAB). This is not me reading research in another field, or me studying some research area I find interesting, it is something essential to what I do.

Where are your proofs?

In order to prove anything, I need to rely on a notation that is consistent with some system. For example, nobody talks about spanning a space in R2 with vectors that don't exist in R2. You do this. I can't prove anything in a system unique to you.

These two links (1 & 2) are more specific to identity matrices and standard bases for the problems with your notation above, you might benefit from a review like this, and although I haven't actually looked at this one 400 pages has to cover a fair amount

directly contradicted the idea that Boolean algebras are field structures

but...The word "field" in "field of sets" is not used with the meaning of field from field theory.

because a field is not a field of sets, not because a Boolean algebra isn't a field:

From Wikipedia

...
The word "field" on the wiki page links to Field (mathematics)

Can you prove that this is a field

I've used or gone over it so many times I can not look at a reference and it would still be entirely someone else's work. But I can't prove anything when you aren't using any known system.

but I don't think the latter is a field

I'm really not concerned with how one particular subset of the integers is a field, but with clarifying what a field is and what a field extension is.

The question relates to whether V is a field.

Did the slithy toves gyre and gimble in the wabe? Do colorlous green ideas sleep furiously? Were the mome raths outgrabe?

The form of a question doesn't mean it is answerable when you are dealing with something undefined. There are infinitely many vector spaces V that can be defined over a field. That's what makes field theory powerful: extension.

It maps two functions to a single function

It doesn't. I gave your multiple scanned or linked images to operations with functions. You came back with something that makes no sense and no references or anything else to show that it meaningfully relates to operation on functions, which is a moot point anyway as I already showed you that operations on functions in high school math differ.

Did you notice that we used two different arrows?

Yes. This

defines a family of functions that map elements from the 2 sets. The pusher is less common, but the above is littered across textbooks through grad school and throughout a variety of journals.

you aware of what "operator overloading" refers to?

I discussed it here.

If you define a plane in R3

Example 1

I say that 2 n-D vectors will span R2.

Then you're usage is unique to your system and you haven't defined one.

 

[LegionOnomaMoi]

Once again, my apologies for having to cut it short (well, my version of short), and I hope the following will end up being mostly compensable as it's been a long day.

This does not prevent us from knowing everything there is to know about the wave, which is what I would have meant by "complete knowledge".

I completely agree. I've been thinking I was following your usage, but I must have made one of my many mistakes in some post and then perpetuated thinking that I was using your formulation. Also, I hope you will forgive the following quotation because it was for me what brought together several varying statements into one idea, albeit still with distinctions:
"note that Postulate 1 states that “information about a quantum-mechanical system is contained in the quantum state.” But could information also reside somewhere other than in the state? Elsewhere you may read statements such as “all information resides in the quantum state,” or “the most complete information resides in the quantum state.” In this view, the quantum state provides a complete description of a quantum mechanical system. Postulate 1, however, leaves open the possibility that the quantum state may not be the whole story—that there could be some deeper layer of physics beyond quantum mechanics.

Spoiler: The 3 sentences that finish the paragraph

These two views thus entail profoundly different philosophical perspectives. Indeed, the debates in foundations of quantum mechanics, still ongoing after 80 years, largely spring from the difference in these perspectives.

"

Bowman, G. E. (2008). Essential Quantum Mechanics (Oxford University Press).

I am not about to try to get into the "two views", but I merely wished to direct attention to the varying ways in which the postulate is stated (which had confused me for a while early on and I believe it was this book which helped connect the dots). It seems to me that even if we accept as a postulate that the QM system's state contains all information about the system, given the way experimental procedures are conducted, this simply means that, if true, we can predict possible outcomes.

Why else would Shankar differentiate the two processes in the way he does (as I addressed in the first part of my response?)

Furthermore, uncertainty principles (classical and quantum) do not prevent us from knowing "the exact state" of the system (emphasis added).

It doesn't prevent us from representing it. The question is whether the representation differs from the bijective correspondence of classical physics, and Shankar seems to me to deliberately point this out.

In QM, the state is described by a wavefunction or ket

Yes. But it is unclear how the representation corresponds to the system. Hence the distinction Shankar makes when comparing classical vs. quantum postulates. In classical mechanics we start out with a system specified by position and velocity, while in QM we obtain an initial state after measurement. Moreover, we require a separate formalism to do this beyond the representation of either the system's state or measurement.

For starters, before we continue, do you acknowledge the distinction between these two cases?

Yes.

I find this confusing. The second sentence is simply not true, by definition. The third sentence seems to contradict the second.

The second sentence was our understanding of physics a century ago when physical systems and physics had achieved so much once we figured out the rather insignificant 2 problems Lord Kelvin referred to, physics would be done. I'm sorry to tell you that he was wrong, and you may want to brace yourself for a shock: the "small problems" fundamentally changed physics and although Max Planck's advisor told him not to go into physics (because, again, what was left to do?) there are still physicists out there. [/flippant response]

The third sentence is what I believe to be true: there is ontological indeterminacy.

Are you suggesting that the ket represents the complete state of an unphysical system?

I think that the most one can say about it is this:
"The state vector Ψ constitutes an unprecedented way of describing nature. It is an abstract entity that carries information about the results of possible measurements. It replaces the classical concepts of position and momentum in the description of physical systems." Bes' Quantum Mechanics: A Modern and Concise Introductory Course, 3rd ed. (Graduate Texts in Physics)."
We have exchanged the physical systems of classical physics and their mathematical representations with mathematical abstractions we refer to as systems. While we know that these abstractions do somehow correspond to systems, our abstract formulation that allows relate to experimental outcomes with remarkable success is not a classical mathematical description of a system. The "entity that carries information" is deliberately and necessarily divorced from any system except itself (the notation), so that we can relate this abstract mathematical state in Hilbert space with experimental outcomes. We do not, however, know how the abstract state corresponds to any ontological reality. Hence Silverman's statement "the wave function exists mathematically". Interestingly enough, one of the most derisive statements I've come across that there is something about QM we don't understand is his:
"Competent physicists, who use quantum mechanics on a daily basis...clearly must understand the instrument with which they are working"

Yet for him QM is irreducibly statistical and the wave function exists only mathematically. I would say that that the standard interpretation is fairly close to this especially in practice, but that (contra Silverman) as we obtaining physical results from measurement QM is more than a formalism for experimental design and does in some way relate to a "real" system.

One would hope that it goes without saying, in a theory of physics described in a physics textbook, that the thing we are trying to represent is physical--an electron, for example, or an atom, etc. By assumption.

I just gave you a quote from a graduate level physics textbook that says this the above is wrong. I can give you a whole lot of references to non-textbook material (i.e., QM literature) that go much further, as textbooks are at least supposed to try not to be partisan in a divided field.

 

[LegionOnomaMoi]

One would hope that it goes without saying, in a theory of physics described in a physics textbook, that the thing we are trying to represent is physical--an electron, for example, or an atom, etc.

One of the reasons that I asked about your specific work and what it entails is because I know how much a textbook can leave out or distort. Obviously, this becomes much less so when one is in a grad program, but for me even here there was a certain amount of understanding gleaned by actually implementing some experimental design, rather than reading about it. Whatever technology or designs you are used to doing, they are clearly physics and when I use the same technology you might I don't deal with it at the level you do. So a bare-bones description of experimental physics in QM was not enough for me, because it was intended to be both simplified and something an "experimental physicist [would] recognize [as] a description of how he uses quantum theory".

For and experimental physicist such as yourself, however, perhaps a bare-bone version is enough as you have the experience I lack to relate it to. If not, simply ignore this post:

"Quantum theory is a procedure by which scientists predict probabilities that measurements of specified kinds will yield results of specified kinds in situations of specified kinds. It is applied in circumstances that are described by saying that a certain physical system is first prepared in a specified manner and is later examined in a specified manner. And this examination, called a measurement, is moreover such that it can yield, or not yield, various possible specified results.

The procedure is this: The specifications A on the manner of preparation of the physical system are first transcribed into a wave function

[hereafter referred to as Ψa (x)]. The variables x are a set of variables that are characteristic of the physical system being prepared. They are called the degrees of freedom of the prepared system. The description of the specifications A is couched in a language that is meaningful to an engineer or laboratory technician. The way in which these operational specifications A are translated into a corresponding wave function Ψa(x) is discussed later.

The specifications B on the subsequent measurement and its possible result are similarly couched in a language that allows a suitably trained technician to set up a measurement of the specified kind and to determine whether the result that occurs is a result of the specified kind. These specifications B on the measurement and its result are transcribed into a wave function

[hereafter referred to as Ψb (y)], where y is a set of variables that are called the degrees of freedom of the measured system.

Next a transformation function U(x; y) is constructed in accordance with certain theoretical rules. This function depends on the type of system that was prepared and on the type of system that was measured, but not on the particular wave functions Ψa(x) and Ψb(y). The “transition amplitude”

<A|B> =

U ( x ; y )

*(y)dxdy

is computed. The predicted probability that a measurement performed in the manner specified by B will yield a result specified by B, if the preparation is performed in the manner specified by A, is given by

P (A,B) = |<A|B>|^2

The experimental physicist will, I hope, recognize in this account a description of how he uses quantum theory. First he transforms his information about the preparation of the system into an initial wave function. Then he applies to it some linear transformation, calculated perhaps from the Schrödinger equation, or perhaps from the S matrix, which converts the initial wave function into a final wave function. This final wave function, which is built on the degrees of freedom of the measured system, is then folded into the wave function corresponding to a possible result. This gives the transition amplitude, which is multiplied by its complex conjugate to give the predicted transition probability.

In a more sophisticated calculation one might use density matrices [notation for these] instead of [notation above for the two transcribed wave functions] to represent the prepared system and the possible result. This would allow for preparations and measurements that correspond to statistical mixtures. But this generalization could be obtained also by simply performing classical averages over various &#936;a(x) and &#936;b(y).

The above account describes how quantum theory is used in practice. The essential points are that attention is focused on some system that is first prepared in a specified manner and later examined in a specified manner. Quantum theory is a procedure for calculating the predicted probability that the specified type of examination will yield some specified result. This predicted probability is the predicted limit of the relative frequency of occurrence of the specified result, as the number of systems prepared and examined in accordance with the specifications goes to infinity.

The wave functions used in these calculations are functions of a set of variables characteristic of the prepared and measured systems. These systems are often microscopic and not directly observable. No wave functions of the preparing and measuring devices enter into the calculation. These devices are described operationally. They are described in terms of things that can be recognized and/or acted upon by technicians. These descriptions refer to the macroscopic properties of the preparing and measuring devices. The crucial question is: How does one determine the transformations: How does one determine the transformations

and

? These transformations transcribe procedural descriptions of the manner in which technicians prepare macroscopic objects, and recognize macroscopic responses, into mathematical functions built on the degrees of freedom of the (microscopic) prepared and measured systems. The problem of constructing this mapping is the famous &#8220;problem of measurement&#8221; in quantum theory."

pp. 53-54 of Stapp, H. Mind, Matter, and Quantum Mechanics (3rd. Ed.) (Springer; 2009)

 

[Mr Spinkles]

I don't think that he [Shankar] says what you did.

I think you're splitting hairs, but to each his own.

Where, in the schema Shankar uses, does he state that the operator is the mathematical representation of measurement?

When I say "measurements are represented by operators", that is intended to be a shorthand description of what Shankar says. E.g., the unmarked equation at the top of p. 124, "The effect of measurement may be represented schematically as follows: ". Or Eq. (4.2.11), a special case where the initial ket before measurement happens to be an eigenket. Clearly, there is a representation of measurement in QM, and it has something to do with operators. On an internet discussion forum, naturally, I prefer to summarize. For greater precision one may consult textbooks, no need to copy them verbatim here.

Obviously, it goes without saying, QM and CM are different. You have articulated some of those differences and I don't deny them.

One point of difference that I seem to notice is that you don't seem to consider |Psi> to correspond to something "real" until an ideal measurement has been performed and it collapses into an eigenket, and a definite position (say) has been obtained. You want X and P to be the real things that describe real systems, and you don't want X and P to be smeared out or undetermined. I may not be doing justice to your position, but in any event, personally I don't feel that way and I don't think a lot of physicists feel that way. To me, an electron in an s-orbital (say) is smeared out in a spherical region around the nucleus. The wavefunction captures that by being itself smeared out. Now, even before we do an ideal position measurement, it's not that we know nothing, I would say. We know the electron is smeared out in its s-shell, analogous (but not equivalent of course) to a classical wave. We know how it's smeared out. And we know how it will respond when disturbed. All these things are certainly weird, and different and less intuitive and more abstract compared to classical mechanics. But there is a certain physical intuition about |Psi> too, IMO. I don't see |Psi> as being completely unphysical, or only predicting future measurements which are real. It's a weird spread-out spherical thing, more or less localized around the nucleus in a particular way, and it has a definite energy and it is not changing in time. It makes sense that |Psi> is that way, since the electron is that way, too--even without doing an ideal position measurement.

When you say

Yes. But it is unclear how the representation corresponds to the system. Hence the distinction Shankar makes when comparing classical vs. quantum postulates. In classical mechanics we start out with a system specified by position and velocity, while in QM we obtain an initial state after measurement.

[emph added] I think it's misleading. In both cases you start out with an initial state, by assumption. In CM that initial state is given by x's and p's. In QM it is not given by x's and p's, but by the initial ket, |Psi, t=0>. In QM you never have a definite x and p simultaneously anyway, before or after measurement. But perhaps I've misunderstood you.

Alas, I have to go to an after funeral gathering and cannot respond to the rest.

I am very sorry for your loss and extend my sincerest condolences.

Your last post is about the "problem of measurement". I think it accurately describes a significant unresolved problem in QM and something that is definitely (even deliberately) left unresolved by the orthodox interpretation. Griffiths acknowledges the same thing. I think that puts it rather well. I also think it is distinct, and not at all contrary to, the much more modest claims I made about |Psi> representing a physical system, etc.

 

[LegionOnomaMoi]

I think you're splitting hairs, but to each his own.

Spoiler: philosophical aside

One attraction I had to the cognitive sciences was the number of interdisciplinary fields that are connected to it, such that one could be a physicist or a philosopher and be working in some brain & behavior lab at Harvard or UCLA, or be an engineer of a psychologist and work at the JHU applied physics lab.

Back when cognitive science was an interdisciplinary field, and not the cognitive sciences (a set of interdisciplinary fields), both philosophers and linguists were founding contributors. And when one is dealing with how one defines meaning itself (something even a computer scientist at the HCI conference may very well have to deal with) splitting hairs becomes essential. We say that Newton (and sometimes we acknowledge Leibniz as well if we think the Germans might see) "invented" calculus, but we tend to not talk about the 200 years until we had a formal definition of limits, still less how incredibly counter-intuitive that definition was compared to the intuitive but (then) ill-defined notion of infinitesimals. And even though infinitesimals were formally defined by Abraham Robinson in the 60s, I know of only one or two textbooks that use his work.

Whereas philosophy has become less than secondary in most sciences, quantum physics is a domain in which one can still find those who reference Plato and Planck, and nobody splits hairs like those who have studied philosophy. Gisin recently contributed to a volume on free will. Stapp & Penrose both have different models of quantum consciousness and both reference philosophers as far back as the Pythagoreans. I have a monograph Interpreting Physics: Language and the Classical Quantum Divide. One thing I have noticed studying philosophy, mathematics, and science developing from Plato to Penrose is that splitting hairs can be a wasted exercise or make all the difference in the world.

In mathematics, even fuzzy set theory is defined precisely. Everything is formal, unambiguous, even elegant. The cosmos, however, has cruelly mocked this art form by making even the most elegant models fail.

E.g., the unmarked equation at the top of p. 124,

Right after he says "This, then, is the big difference between classical and quantum mechanics: an ideal measurement of any invariable &#969; in classical mechanics leaves any state invariant, whereas the ideal measurement of &#937; in quantum mechanics leaves only the eigenstates of &#937; invariant."

Clearly, there is a representation of measurement in QM, and it has something to do with operators.

"Almost all treatments of quantum mechanics agree in ascribing fundamental importance to the notions of "state" and "observable". The physical interpretation of these notions, however, differs considerably from one author to another"
(1.1 States and Effects)

I agree completely with what you said above. I just think that the difference between representations in QM and classical physics is more than simply splitting hairs (or if it is not, they are some pretty important hairs).

On an internet discussion forum, naturally, I prefer to summarize.

Naturally, anybody would and should. I just suck at it.

For greater precision one may consult textbooks, no need to copy them verbatim here.

I didn't do that just to fill in your gaps but because I recently learned (and have Polyhedral to thank) about LaTex and how to make mathematical symbols into linkable images without scanning pages or filling up my allotment uploaded of images here. So it was both practice and it allowed us to look at the same mathematical representations.

Obviously, it goes without saying, QM and CM are different.

Of course. It seems we disagree on the nature of those differences and/or their importance.

One point of difference that I seem to notice is that you don't seem to consider |Psi> to correspond to something "real" until an ideal measurement has been performed and it collapses into an eigenket, and a definite position (say) has been obtained.

That's not actually my view, although it is a good summary of the view I have expressed. It is, rather, my view that such a description comes closest to the standard or orthodox interpretation:

"Niels Bohr brainwashed a whole generation of physicists into believing that the problem (of the interpretation of quantum mechanics) had been solved fifty years ago." (Murray Gell-Mann, Noble Prize acceptance speech, 1976)

Spoiler: A description by someone else at the speech

"I vividly recall the occasion of a lecture on the measurement problem given in the early 1970s at The Rockefeller University by a Nobel laureate in physics. The reaction of the audience, composed largely of theoretical physicists and mathematicians, was distinctly cool if not unfriendly. The skepticism was directed not so much at the proposed solution as to the notion that there was a problem to be solved. After the lecture, the laureate remarked ruefully: "I suppose that I will have to do something new to restore my reputation." Today his lecture would likely get a different reception...On the one hand, the problem can be seen as revealing that there is something rotten at the core of the theory because of its inability to give a satisfactory description of what occurs in the interaction between an object system and a measurement apparatus."

Earman's Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes (Oxford University Press; 1995)

you don't want X and P to be smeared out or undetermined.

I'm perfectly happy with indeterminacy and it is in fact my view. I simply think that the difference between correspondence schemata of classical and quantum physics are qualitatively different, and that the formalisms reflect this.

To me, an electron in an s-orbital (say) is smeared out in a spherical region around the nucleus.

I dislike the standard wave-particle duality description, but rather side with M. Y. Han here (A Short Story of Light): "an important caveat is in order about a matter of terminology in quantum physics. The new quantum reality, the wave–particle duality, describes a quantum thing that is neither a particle in classical sense nor a wave in classical sense. We can shorten the name to simply duality, that is, electrons, protons, neutrons and photons, etc. should all be called duality, certainly not particle nor wave"

To me, everything is nonlocalized, but most things are so localized that the difference is impossible to measure let alone matter (no pun intended). All matter is waves that differ in how localized they are, and once we begin to leave the subatomic realm, they tend to get very localized very quickly.

I don't see |Psi> as being completely unphysical, or only predicting future measurements which are real.

I agree.

I think it's misleading. In both cases you start out with an initial state, by assumption.

True. However, in the QM case we don't describe or represent that initial state until we have already made measurements s.t. we have prepared a quantum system through an ensemble but which is not an ensemble (in the standard interpretation, although we do prepare an ensemble, it is not interpreted as one in the way that it is in an ensemble interpretation). So the initial state in QM is given by a very different kind of preparation, transcriptions, specification, etc., process.

But perhaps I've misunderstood you.

Or I'm wrong. Both are likely, as I am not good at conveying things well and I am not a physicist.

I am very sorry for your loss and extend my sincerest condolences.

Thank you. I appreciate that very much. And thanks again for bearing with me here.

 

[Mr Spinkles]

Legion,

On the splitting hairs thing and "specified" vs. "represented" .... well, yes, of course, there are differences between CM and QM. As always. I guess a concern I have is that not only are you splitting hairs, but you're splitting them always in a particular direction, against the very mild claims I made, rather than in favor. To wit, I repeatedly made a statement about |Psi> being the mathematical representation of the state of a system. You objected in all sorts of ways to that statement, bringing up delta functions and expectation values and density matrices, until you saw my statement repeated almost verbatim by Shankar. Then you decided, actually, the statement about |Psi> representing a QM state is okay. It's the other statement I made about CM which you were objecting to all along!

(BTW I noticed that when I quoted Shankar I erroneously wrote "represented" instead of "specified" in CM postulate I. That was a typo. If I had noticed different wording I would not have intentionally changed it, of course.)

It was the same thing with Griffiths. By the rules of plain English, Griffiths stated in the parts you yourself quoted that the wave function represents the state of a particle. But, you argue, the wave function does that via the statistical interpretation! Okay. That's additional info. about how the wave function represents the state of a particle. But clearly, Griffiths says the wave function represents the state of a particle. I mean on p. 433 he says, and I quote, "The wave function (Psi) represents the state of a particle" and then he says this interpretation is "shared by most physicists today". If that's not exactly, word-for-word what I've said all along, it's gosh darn close. How can you skip over stuff like that without acknowledging that perhaps your objections to my bold, italicized statements about |Psi> (and Griffiths!) were hasty? Am I presumptuous in expecting some kind of mea culpa?

Right after he says "This, then, is the big difference between classical and quantum mechanics: an ideal measurement of any invariable &#969; in classical mechanics leaves any state invariant, whereas the ideal measurement of &#937; in quantum mechanics leaves only the eigenstates of &#937; invariant."

Yes, in QM it's an ideal "measurement" of Omega, a (Hermitian) "operator" which "represents" the property being measured. Or, as Griffiths put it, "Observables are represented by hermitian operators". Or, as I put it, "Measurements are represented by operators acting on kets..." No surprises here.

I agree completely with what you said above. I simply think that the difference between representations in QM and classical physics is more than simply splitting hairs (or if it is not, they are some pretty important hairs).

Well yes, that's true.

That's not actually my view, although it is a good summary of the view I have expressed. It is, rather, my view that such a description comes closest to the standard or orthodox interpretation:

"Niels Bohr brainwashed a whole generation of physicists into believing that the problem (of the interpretation of quantum mechanics) had been solved fifty years ago." (Murray Gell-Mann, Noble Prize acceptance speech, 1976)

Spoiler: A description by someone else at the speech

"I vividly recall the occasion of a lecture on the measurement problem given in the early 1970s at The Rockefeller University by a Nobel laureate in physics. The reaction of the audience, composed largely of theoretical physicists and mathematicians, was distinctly cool if not unfriendly. The skepticism was directed not so much at the proposed solution as to the notion that there was a problem to be solved. After the lecture, the laureate remarked ruefully: "I suppose that I will have to do something new to restore my reputation." Today his lecture would likely get a different reception...On the one hand, the problem can be seen as revealing that there is something rotten at the core of the theory because of its inability to give a satisfactory description of what occurs in the interaction between an object system and a measurement apparatus."

Earman's Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes (Oxford University Press; 1995)

I think those who favor the orthodox interpretation recognize that it has limitations. Griffiths stated it pretty convincingly. The argument in favor of it is simply that it is the least bad interpretation we have so far--it corrals the conceptual difficulties of QM into a relatively small (but of course significant) area, which is the measurement problem. In other words, most physicists simply find it to be the most useful, if imperfect, interpretation.

One area we it seems like we disagree, is that IMO, in the orthodox interpretation, if you ignore these limitations (namely the measurement problem), you are left with a pretty convincing and intuitive connection between the mathematical representation on the one hand, and the physical reality OTOH. Yes it's certainly more abstract than classical mechanics, for all the reasons you cited (statistical interpretation, etc.) No doubt about it. But I would not go so far as to say that |Psi> is just a mathematical trick to churn out results of ideal measurements, or even that that's all we can say. (I hasten to add, that is not to say that you have gone that far.) Obviously that could turn out to be the truth one day, but at the present moment that seems to describe the agnostic interpretation, or some interpretation where QM is incomplete.

The orthodox interpretation, as I have always understood it, is the one that takes QM more or less "literally" (and Griffiths seems to agree). When I imagine the wavefunction, Psi(x,y,z), or let's say the probability density, |Psi(x,y,z)|^2, under the orthodox interpretation, I really do think of that as representing a quantum particle, just as the position and momentum coordinates represent a classical particle. Of course, the representation and the way it behaves are different in the quantum vs. the classical case. This reflects the fact that a quantum particle and the way it behaves is different from a classical particle. When I say the representation is analogous I am only saying that the |Psi> is just as "real", and no more of a trick or device, than the classical coordinates.

Let me unpack that statement with specifics. Here's what I'm thinking of, specifically: a quantum particle is spread out in some particular way in space, and tends to occupy certain areas of space more than others; so it is with Psi. IOW it doesn't just help you calculate the possible results of future ideal measurements; it also tells you something about the particle right now, namely, that it is spread out and evolving (via Schrodinger's equation) in a particular way, and therefore, so are its properties (such as position). So of course, Psi reflects that. Furthermore, a quantum particle is forced into having a particular value of a certain property, such as spin, when an ideal measurement is performed, and this occurs in a statistical way; and so it is with Psi. After an ideal position measurement, a quantum particle is strongly localized at the position obtained in the measurement and immediately begins spreading out again according to a particular deterministic law; so it is with Psi.

Think of those pictures of various electron orbitals around molecules that we have all seen in chemistry. Or, think of those atomic force microscopy images of "quantum corrals" containing "ripples" of electrons, or the "hologram" of a single atom occupying two positions at once. These are ultimately reconstructions of the wavefunction (or probability density), as I'm sure you're well aware. I find it quite useful, and commonplace, to say "that's an accurate representation of the particle, right now". Not just some clever device for calculating the possible results of future ideal measurements. Obviously, the pictorial representation of such a weird mathematical/physical object has limitations, but it's also hard to represent curved spacetime pictorially. The point is, that like curved spacetime, the mathematical representation corresponds accurately to the underlying physical reality, however weird it may be. It's not just some device or trick that happens to work.

But it seems we disagree that this could be called the "orthodox" or most-common interpretation of QM. The quotes you gave above, however, do not seem to resolve this disagreement (or perhaps confusion).

To me, everything is nonlocalized, but most things are so localized that the difference is impossible to measure let alone matter (no pun intended). All matter is waves that differ in how localized they are, and once we begin to leave the subatomic realm, they tend to get very localized very quickly.

If only we had a mathematical representation, which would also be a nonlocalized wave, which tended to get localized at the macroscopic level. Then the representation and the particle would correspond one-to-one.

True. However, in the QM case we don't describe or represent that initial state until we have already made measurements s.t. we have prepared a quantum system through an ensemble but which is not an ensemble (in the standard interpretation, although we do prepare an ensemble, it is not interpreted as one in the way that it is in an ensemble interpretation). So the initial state in QM is given by a very different kind of preparation, transcriptions, specification, etc., process.

I don't quite understand you here. Is this about the density matrix again?

 

[LegionOnomaMoi]

On the splitting hairs thing and "specified" vs. "represented"

YOU'RE the one who chose to be a member of the elite scientists who believe that every time Paris Hilton deals with split ends, universes divide.

I repeatedly made a statement about |Psi> being the mathematical representation of the state of a system. You objected in all sorts of ways to that statement, bringing up delta functions and expectation values and density matrices, until you saw my statement repeated almost verbatim by Shankar.

Verbatim?

QM postulates the existence of a ket which represents the complete state of a physical system, not just our knowledge of it, and this is the standard, most common way of formulating QM

Shankar states (emphasis added): "As far as the state vector |&#936;> is concerned, there is just one space, the Hilbert space, in which it resides", p. 121.

Where, exactly, is this physical Hilbert space? When did a mathematical space become not just a representation of physical reality but a place for physical systems to reside (Hint: p. 67 of Shankar, in which "physical" Hilbert space is a function space which differs from the mathematical version "which contains only proper vectors....The role of improper vectors in quantum theory will be clear later")?

Nor is this true:

You objected in all sorts of ways to that statement, bringing up delta functions and expectation values and density matrices

You brought up density operators

For example, your first reference Theoretical Foundations of Quantum Information Processing and Communication is talking about the use of a density operator.

The entire paper I cited concerns interpreting QM in terms of probability theory. Nor did I bring up Dirac/delta functions. You did that when you made (contra Shankar) yet another statement about the way in which "quantum statistics" differs from quantum mechanics (emphasis added)

I am guessing you are wondering how we achieve the latter, without resorting to the approximation of quantum statistics (using density operators)? There are many ways, but one example: perform an ideal measurement on the position 'x' of an electron. Then the wavefunction Psi(x) of that individual electron immediately after measurement is a delta function centered at 'x'.

So what do we do without resorting to this statistical aid such that we can't "interpret

as the probability for finding the particle with value &#969; for &#937;" because "the number of possible values for &#969; is infinite and the total probability is unity, [so] each single value of &#969; can be assigned only an infinitesimal probability" (p. 121)? Why does Shankar (p. 125), say "Quantum theory, on the other hand, makes statistical predictions about a particle in a state |&#936;>"?
How do we have complete knowledge about an arbitrary state of a physical system ("we begin with an arbitrary state |&#936;>" p. 125) which exists as a mathematical abstraction that is related to physical reality only upon measurement? And not only that, but a measurement that "tells us' nothing compared to the complete specifications of the state just after measurement" (italics in original, emphasis added)?
When you can explain how physical systems reside in Hilbert space, then we can worry about what "statistical predictions" means to Shankar. Until then, QM postulates the state of a mathematical abstraction in an abstract mathematical space.

That's additional info. about how the wave function represents the state of a particle.

Which is what I said from the beginning:

The correspondence I refer to is this:
"In classical physics, the notion of the “state” of a physical system is quite intuitive...there exists a one-to-one correspondence between the physical properties of the object (and thus the entities of the physical world) and their formal and mathematical representation in the theory...With the advent of quantum theory in the early twentieth century, this straightforward bijectivism between the physical world and its mathematical representation in the theory came to a sudden end. Instead of describing the state of a physical system by means of intuitive symbols that corresponded directly to the “objectively existing” physical properties of our experience, in quantum mechanics we have at our disposal only an abstract quantum state that is defined as a vector (or, more generally, as a ray) in a similarly abstract Hilbert vector space."
The quote continues, but the rest is contained in PolyHedral's response

Like Shankar, once again we find that QM postulates a mathematical entity residing in a mathematical space used to make statistical predictions in the physical world.

I mean on p. 433 he says, and I quote, "The wave function (Psi) represents the state of a particle" and then he says this interpretation is "shared by most physicists today". If that's not exactly, word-for-word what I've said all along, it's gosh darn close.

It would be, had you not said added the stuff about "complete state" and "physical" when both your texts refer to probabilities and abstract mathematical entities.

How can you skip over stuff like that without acknowledging that perhaps your objections to my bold, italicized statements about |Psi> (and Griffiths!) were hasty?

Because I don't base my understanding of QM on two textbooks and three courses but 2+ years of research on quantum physics and several hundred volumes and many more papers, so I know the difference between the state of a physical system and the state of a quantum system when you assume the two are the same. I may be dumb, but I'm not such an idiot (I hope) that I somehow missed the repeated references to representation of states. I just knew that these representations didn't have the one-to-one correspondence with any physical system, but you seem to have assumed this. You might try going back over your statements and your two textbooks keeping in mind the possibility that system does not refer to a physical system, and that the representation of quantum systems vs. the representation of classical systems (as I said from the start of our conversation about this) is not at all equivalent. The latter is the bijective correspondence while the former is a "who the **** knows".

 

[idav]

When it is said that everything is "waves differing on how localized they are" can't be because of the evidence that photons are particles. The photoelectric effect shows that light can't be just a wave and it isn't infinitely divisible.

 

[LegionOnomaMoi]

When it is said that everything is "waves differing on how localized they are" can't be because of the evidence that photons are particles. The photoelectric effect shows that light can't be just a wave and it isn't infinitely divisible.

The photo-electric effect combined with other experiments (esp. Young's) shows that the entire conceptions of particles and waves were both wrong. All we have are entities like electrons and photons that vary in how localized they are. Also, to date it is not settled whether infinite divisibility a property of matter, time, or spacetime.

 

[idav]

The photo-electric effect combined with other experiments (esp. Young's) shows that the entire conceptions of particles and waves were both wrong. All we have are entities like electrons and photons that vary in how localized they are. Also, to date it is not settled whether infinite divisibility a property of matter, time, or spacetime.

Your just insisting there is a collapse when we can't really prove that as the experiments rule out the smeared out or spread out photon. Experiments confirm a particle with wave properties unless you can prove the photon wasn't really there at the point of observation.

 

[PolyHedral]

Your just insisting there is a collapse when we can't really prove that as the experiments rule out the smeared out or spread out photon. Experiments confirm a particle with wave properties unless you can prove the photon wasn't really there at the point of observation.

Counterfactual measurements?

 

[idav]

Counterfactual measurements?

That'll work.

 

[LegionOnomaMoi]

Your just insisting there is a collapse when we can't really prove that as the experiments rule out the smeared out or spread out photon.

First, I'm not insisting there is a collapse. That's just a word used to define why our complete dynamical description of the photon or electron doesn't actually give us deterministic results. It's how we explain why the mathematics that work so well at predicting experimental outcomes if we model the system as if it were a wave, but we always get a kind of localized result.

Second, the experiments prove nothing of the sort. A "smeared out" particle is a way of referring to the space taken up by some particle that is greater than the space taken up of what we observe. When shine a flashlight in a dark room, you may see most of a painting, but not all of it because your flashlight is too small to cover the area of the painting. Does that mean the painting is smeared out? No.

You are claiming we can show that an electron or photon or whatever is not smeared out, but you can't tell me how much area it is "supposed" to cover. There' no detection you can point to and say "it's not smeared out" because all you've done is shown that we've detected a certain amount of something. You can shine a flashlight or a laser or a lamp and you will see light that covers different amounts of area on a wall. Until you can tell me how much area a "particle" of light takes up, you can't tell me we have ever shown it isn't smeared out.

Experiments confirm a particle with wave properties unless you can prove the photon wasn't really there at the point of observation.

You keep saying things like this as if they make any sense. It's like saying "experiments confirm dead people have living properties" or "experiments confirm a dog with cat properties". It's meaningless.

All you can do is show me that e.g., a light can be detected in some area. It always could. In order to say it is a particle or has particle properties you need more than just detection in a localized area, because the light from a flashlight is more localized than a lamp and a laser is detected as more localized than both. We don't need QM to tell us that light isn't detected as "smeared out" over infinite space. We need it to explain why even when we send discrete units of light at the splitting screen, we can never explain why they show up where they do unless we say treat them like waves.

 

[LegionOnomaMoi]

I posited by assumption the existence of a physical system which does not obey deterministic laws of evolution.

I can't believe I didn't think of this earlier. Let's say the above is so. Why do we have the Schrödinger equation (the deterministic, or time-dependent, formulation, as found in Griffiths' text on p. 1 and in Shankar's on p. 116)? Given an initial fixed state of a generalized quantum system, the state vector is completely defined at any future time t and thus evolves in an entirely deterministic manner. So why do we have a model for quantum systems that is completely deterministic if quantum systems are really, truly, not deterministic? [he asked rhetorically]

 

[Mr Spinkles]

Legion,

I may not be able to get back to you for a while ... I had written a response, almost finished, and then my power went out. It was like straight out of an Alanis Morissette song, lol.

I posited by assumption the existence of a physical system which does not obey deterministic laws of evolution.

I can't believe I didn't think of this earlier. Let's say the above is so. Why do we have the Schrödinger equation (the deterministic, or time-dependent, formulation, as found in Griffiths' text on p. 1 and in Shankar's on p. 116)? Given an initial fixed state of a generalized quantum system, the state vector is completely defined at any future time t and thus evolves in an entirely deterministic manner. So why do we have a model for quantum systems that is completely deterministic if quantum systems are really, truly, not deterministic? [he asked rhetorically]

Yes, I'm aware Schrodinger's equation is a deterministic law of evolution. Notice that I did not insist the system must always "not obey deterministic laws". As long as we assume the system "does not obey deterministic laws" at least some of the time, i.e., it does not completely obey deterministic laws, that is sufficient for my argument. And in context, that is all I meant. (To wit, in QM the deterministic Schrodinger equation is occasionally interrupted by a non-deterministic collapse upon measurement, as we both know.) With that clarification in mind, here's the exchange, in context (emphasis added):

If it is a physical system, then complete knowledge of the state entails the ability (in principle) to completely predict the systems evolution. We frequently say complex systems are indeterministic, but we mean they are deterministic in principle, but in practice we are unable to determine their dynamics.

If we can't, even in principle, describe the evolution of a physical system, then either our knowledge of the system's state is incomplete, QM is incomplete, or (the most widely accepted view) the state of the system doesn't correspond to a physical system at all, merely a mathematical device to tell us given preparation a and measurement b, we'll get outcome x with y probability.

The first sentence in bold is clearly a non-sequitur. Otherwise you've simply defined out of existence a physically real, indeterminate system! That begs the question. I posited by assumption the existence of a physical system which does not obey deterministic laws of evolution. When we entertain this possibility, rather than rejecting it by assumption, there is a fourth option in your second paragraph: the physical system does not have a predestined evolution, and hence, naturally, our complete mathematical representation also does not have one. Or, as Griffiths put it fancifully somewhere, it's not that we don't know--God doesn't know. (BTW you switched from "predict" in the 1st para. to "describe" in the 2nd, I'm going to assume we can stick with "predict", since that is what actually captures the difference between a deterministic vs. non-deterministic evolution.)

In hindsight I maintain what I said, you were begging the question here. Which is surprising because in your previous post (before the most recent one) you said you believe ontological indeterminacy to be true. I submit to you that if you would aim your extremely sensitive error-detection senses at your own statements, as often as you aim at mine, you might find that you didn't say what you intended and I may even have a point sometimes.

 

[idav]

First, I'm not insisting there is a collapse. That's just a word used to define why our complete dynamical description of the photon or electron doesn't actually give us deterministic results. It's how we explain why the mathematics that work so well at predicting experimental outcomes if we model the system as if it were a wave, but we always get a kind of localized result.

Second, the experiments prove nothing of the sort. A "smeared out" particle is a way of referring to the space taken up by some particle that is greater than the space taken up of what we observe. When shine a flashlight in a dark room, you may see most of a painting, but not all of it because your flashlight is too small to cover the area of the painting. Does that mean the painting is smeared out? No.

You are claiming we can show that an electron or photon or whatever is not smeared out, but you can't tell me how much area it is "supposed" to cover. There' no detection you can point to and say "it's not smeared out" because all you've done is shown that we've detected a certain amount of something. You can shine a flashlight or a laser or a lamp and you will see light that covers different amounts of area on a wall. Until you can tell me how much area a "particle" of light takes up, you can't tell me we have ever shown it isn't smeared out.



You keep saying things like this as if they make any sense. It's like saying "experiments confirm dead people have living properties" or "experiments confirm a dog with cat properties". It's meaningless.

All you can do is show me that e.g., a light can be detected in some area. It always could. In order to say it is a particle or has particle properties you need more than just detection in a localized area, because the light from a flashlight is more localized than a lamp and a laser is detected as more localized than both. We don't need QM to tell us that light isn't detected as "smeared out" over infinite space. We need it to explain why even when we send discrete units of light at the splitting screen, we can never explain why they show up where they do unless we say treat them like waves.

I've already shown that we can detect the range of space a wave takes up.

Anyone has yet to show how a photon is different when going through one slit vs two slits. We get a photon, a small packet of energy, on the screen when sending it through one slit. What changes when it goes through two slits? The photon didn't change just it's trajectory probability changes.

 

[LegionOnomaMoi]

I had written a response, almost finished, and then my power went out. It was like straight out of an Alanis Morissette song, lol.

I hate that. But I don't know what's worse:
1) that it happened, as I know how incredibly irritating it is to see one's work vanish
2) that you are familiar with Alanis Morissette's songs
3) that I'm mocking this even though I have all of jagged little pill on my mp3 player

I did not insist the system must always "not obey deterministic laws".

Yet (with some additional tuning) we can always apply a version of this equation. I found Dirac's The Principles of Quantum Mechanics 4th ed. here. Skipping over the technical nuances: Sec. 12 The general physical interpretation

"In classical mechanics an observable always, as we say, 'has a value' for any particular state of the system. What is there in quantum mechanics corresponding to this? If we take any observable and any two states x and y, corresponding to the vectors <x| and |y>, then we can form the number <x|&#958;|y>. This number is not very closely analogous to the value which an observable can 'have', in the classical theory, for three reasons, namely, (i) it refers to two states of the system, while the classical value always refers to one, (ii) it is in general not a real number, and (iii) it is not uniquely determined by the observable and the states, since the vectors <x| and |y> contain arbitrary numerical factors...In the general case we cannot speak of an observable having a value for a particular state, but we can speak of its having an average value for the state. We can go further and speak of the probability of its having any specified value for the state, meaning the probability of this specified value being obtained when one makes a measurement of the observable." (emphasis added)

We use the deterministic equation regardless of measurement collapse because it is built into this formalism via another: the "jump" to collapsed state. The evolution of the system is mathematically represented as deterministic because the mathematical representation doesn't bijectively map onto the physical system.
That's why even with both the observable and the states, we have no unique outcome. If the operators were merely representations of measurements, then why is it not the case that both the states and the observable give us a value? Because the operator is not a mathematical scheme for measurement. Instead, another layer of formalism is added. Measurement is combined with this theoretically derived formalism s.t. an entirely deterministic "system" has probabilistic results. The projection, collapse, etc., are said to "cause" some change in the system upon measurement, but the obtained values require more than a representation of the system and of measurement.

As long as we assume the system "does not obey deterministic laws" at least some of the time,

This is missing my (poorly made) point. The deterministic mathematical schema of the system remains deterministic even though the outcomes are almost always probabilistic. The only possible way for this to be so is for the representations to correspond to something other than system/measurement (e.g., probable outcomes, information not in the ket, etc.). Otherwise, our deterministic representation would corresponded directly to a physical system in a way it never does.

Which is surprising because in your previous post (before the most recent one) you said you believe ontological indeterminacy to be true.

I do. That's why when Shankar (p. 146-7) speaks of an operator acting on a dynamical system represented by a ket "as a 'rotation' in Hilbert space", it makes sense to me. If I had to interpret a ket as a representation of a system with a one-to-one correspondence, I'd be scratching my head wondering what on earth it means to represent a physical system as rotating in a complex mathematical space with an inner product. Knowing how the language of classical physics was used and passed on by Einstein, Bohr, Schrödinger, and Planck to the next generation (Bohm, Heisenberg, Dirac, and von Neumann), s.t. the terms used to refer to one thing ("state", "system", etc.) remained even when what they represented changed, it makes sense. The postulated ket does indeed contain the information of a system within it, but it cannot possess the analogous bijectivism of classical systems with their mathematical descriptions. This is not a byproduct of indeterminacy, as we can have a deterministic representation without a determined result. It's because the measurement process doesn't fit into your dichotomous correspondence schema between operators-observables.

Shankar, like Dirac and most, tells us the QM analogue to the state in classical mechanics (pp. 117-118). The operator &#937; is constructed after measurement ("given that the particle is in state |&#968;>"). Even better (p. 119), "we have a single ket |&#968;> representing the state of the particle in Hilbert space, and it contains the statistical prediction for all observables. To extract this information for any observable, we must determine the eigenbasis of the corresponding operator and find the projection of |&#968;> along all its eigenkets."

The "statistical predictions" is equivalent to statements like "all the information" or "everything there is to know". We cannot construct the operator (the mathematical function which gives us, or informs us of, an observable) without knowing whether the "particle" is in state |&#968;>. The only way to know that is to have either already prepared an ensemble "system" or to have already measured or both. The point is that we require knowledge of the state obtained through physical interaction with it to construct the operators that correspond to observables.

If you look at Shankar's comparison on p. 115-116 to compare postulates 2 & 3, note that schematically the classical operators correspond to what we get by applying the quantum (Hermitian) operators. This is unrelated to the notation used to represent the quantum system/state, and thus can't be treated by conceptualizing the system as ontologically indeterminate s.t. statements about probabilities or statistical knowledge are not counter-examples to a ket containing all there is to know. We require additional formalism, depicted by Shankar conveniently by using the same notations side-by-side s.t. the uppercase Greek letters in the quantum postulates give us the lowercase Greek letters (technically, majuscule and minuscule). The lowercase Greek operators are in both, not the uppercase.

The question, then, is why we have two operators for a quantum system to get the classical variables? Which one corresponds to (or represents) measurement? Why does postulate III tell us that "If the particle is in state |&#968;>", then we can refer to what the "variable corresponding to &#937;" may yield? If &#937; corresponds directly to observable properties, then this makes no sense. It does as a probability function that, given a particular state, applying the operator will yield one of many possible values. We require two operators because we require a mathematical function that is derived theoretically and that is used in addition to the measurement in order to relate the measurement to some property of the system, and not to correspond to the measurement.

I submit to you that if you would aim your extremely sensitive error-detection senses at your own statements, as often as you aim at mine, you might find that you didn't say what you intended and I may even have a point sometimes.

First, I always believe you have a point (and I generally assume that if I disagree I missed something). Second, you had the advantage of classes and teachers combined with your experience, while I had lots and lots of textbooks, volumes, papers, etc., instead (from grad textbooks to the "historical reader" put together by Wheeler and Zurek). The reason so many textbooks exist isn't because each offers something none of the others do (quite the contrary). But when one does not have a professor to speak to, scrutinizing multiple texts, doing problem sets, and constantly questioning myself over and over and over again is the best I substitute I could manage.
I can still be wrong, of course, but had I not already spent a very long time doing exactly what you suggest, I'd have taken your word for it from the get-go. Recall that I have never once questioned your credentials nor your knowledge, and that you know quite well I have done this when I did not think someone was presenting themselves honestly. You are clearly competent and I've seen you deal with outlandish claims and aggressive responses with a cool head and well-informed replies. If I am wrong, it isn't because I haven't turned the microscope around on me. I do that constantly as is, and had I not continuously done this for some time now, I wouldn't have questioned your position without at least until another hundred texts and many rounds of playing devil's advocate with myself.