who have understood the notation without issue.
How? Or why? After all, if you want to treat the wave function in so basic a sense, you've already introduced more than you need: "In quantum mechanics, the state of a particle is described by a complex-valued function of position and time,
ψ(x, t),
x ∈
R³,
t ∈ R."
Mathematical Concepts of Quantum Mechanics (Springer, 2003). However, the reason this is totally inadequate is apparent from something as simple as the double slit experiment, where
ψ=ψ1+ψ2. And as this is only a simple combination of wavefunctions, in order to generalize the wave function to be applicable to any given quantum system, we require that wave function represent "the state of all possible states of a possible of a particle at a given time" (ibid). Also, the state of a microscopic system "can be loosely expressed by saying that
if a system can exist in different configurations (corresponding for example to different classical descriptions), it can also exist in a superposition of these configurations, so to speak suspended between them" (italics in original). from
Exploring the Quantum (
Oxford Graduate Texts).
Perhaps clearest of all (italics in original; emphasis added): "Moreover, quantum states are not simply specifications of the corresponding classical quantities. Consider a single
classical particle. The state of a classical particle at some time is given by the three position coordinates and the three velocity components of the particle. It turns out, however, that the
quantum state of a single particle at a particular time is, in general, a
function in space.
This means that, in contrast to the classical state, we cant simply specify unique coordinates and velocities of a quantum particle." from Bowman's
Essential Quantum Mechanics (Oxford University Press, 2008)
I think you mean a vector in that space. A single state is represented by a single point in Hilbert space
This ignores the most basic principles (esp. the superposition principle) of QM (emphasis added): "The state of a physical system at
t=t0 is defined in terms of a ket, or a row vector |
ψ0⟩ belonging to the vector space of states.
The space of states is a vector space, therefore it follows that the superposition of two states is again a state of the system. The space of states also contains the concept of the scalar product." from Razavy's
Heisenberg's Quantum Mechanics (World Scientific, 2011). Second, as for "
in that space", again from
Mathematical Concepts of Quantum Mechanics: "the space of quantum-mechanical states of a system is a vector space with an inner-product (in fact a Hilbert space)."
Also, from
Exploring the Quantum (emphasis added): "In laymans language, we may say that
the wave function describes the state of the particle suspended, before measurement, in a continuous superposition of an infinite number of possible positions." Basically, the essence of quantum mechanics (and a main reason it exists, distinguished from classical), is captured by the fact that the state of a quantum system is described by vector
s, plural, making the state (or ket) a vector space. Additionally, not only is an entirely seperate formalism needed (observable operators) to get any information about final states, but also we the accompanying measurement forces what cannot be described in terms of specific, unique coordinates even in Hilbert space to become not "points" or coordinates in Hilbert space, but in R3.
Finally, to include time in the picture at all, we either use it as a parameter or an observable, but never to characterize a quantum system (emphasis added): "
the quantum state of a system at a given time is capable of precisely specifying only a corresponding subset of the physical magnitudes, that is, the currently observed quantities and the others that are functions only of them, in the intervals between measurement events,
this being so only as long as quantities other than these remain unmeasured, that is, 'unobserved.'" from Jaeger's
Entanglement, Information, and the Interpretation of Quantum Mechanics (Springer, 2009).
It parametrizes the line in Hilbert space. I think
It can't. Because no coordinate in Hilbert space can denote time, which is an element of R.
Since time evolution's only parameter is the wavefunction, what else would tell us?
The obervable operator used, which is how we are able to describe the quantum at all. That's often called the "second postulate" of quantum mechanics (italics in original): "Properties of a quantum-mechanical system that can (in principle) be observed, or measured, are called
observables, and are represented by Hermitian operators."
Essential Quantum Mechanics
However, I contend these are flawed interpretations, not experimental verification of any sort.
So the superposition of macroscopic systems (observed experimentally in e.g., "
Quantum interference of large organic molecules") is explainable locally through...? Proof cannot be empirical. And there are many proofs that either QM or GR (or both) entails superluminal signals, nonlocality, and/or causality violations. These are
necessarily mathematical, as (apart from the definition of proof itself) with QM that's what we have (mathematics), and with general relativity we are quite limited in our ability to experiment with CTCs other than mathematically.
This is not to say we don't have experimental evidence of both nonlocality and superluminal signalling. For the latter, see e.g.,
"
Superluminal Twin Beams, Superluminal Images and the Arrival Time of Spatial Information in Optical Pulses with Negative Group Velocity"
"
Superluminal Images and the Arrival of Spatial Information in Optical Pulses with Negative Group Velocity"
"
Stimulated Generation of Superluminal Light Pulses via Four-Wave Mixing" (full text available for free
here)
We find articles like this in journals like
Science and
Nature: "
Quantum teleportation and entanglement distribution over 100-kilometre free-space channels"
In the above article, the authors note: "In comparison to these previous studies, the experiment presented in this Letter
achieves long-distance free-space teleportation of an independent quantum state, thus paving the way for satellite-based global quantum communication."
It confuses inferred observation with actual effect.
The entirety of quantum theory rests on such inference. Hence the problems.
There's a crucial difference - the former is absurd, the latter is sensible.
When a journal like
Science publishes a study like "
Entangling Macroscopic Diamonds at Room Temperature", and when
Nature publishes study like "
Real-time single-molecule imaging of quantum interference" in which the authors state directly that their results are an "unambiguous demonstration" which is "only explicable in quantum terms" and "provide
clear and tangible evidence of the quantum behaviour of large molecules", you'll forgive me if I tend to take such results seriously, rather than evidence that the most respected science journals in the world got really sloppy.
For someone who has so much faith in the possibility of some future experimental evidence that consciousness is possible using a computer, you sure are skeptical of actual experimental evidence when it comes to modern physics.
Quantum teleportation explicitly requires a classical, i.e. subluminal, channel of communication.
The classical requirement is simply that it needs to be capable of human observation: "Quantum teleportation relies on using both a quantum channel and a classical channel between two parties...The quantum channel is used by Alice and Bob to share the entangled auxiliary state." (from "Quantum teleportation over 143 kilometres using active feed-forward")
How would you even tell? Many worlds divergence is, by definition, impossible to observe.
Yes, but it is supposed to tell us why we can't measure the superposition state, or why we can only get a single state for the same system rather than observe the wave-particle (nonlocal) duality. As we can actually observe these now, what purpose does a many-worlds interpretation serve?
"Measuring" a superposition collapses it and produces one answer.
Not anymore. We can now get two answers at once, hence experiments like the one in Science: "A Quantum Delayed-Choice Experiment"
I don't what else Descartes said. I was only quoting one argument.
That argument was about proving consciousness. It isn't something else he said. That was it.
