Quantum physics requires imaginary numbers to explain reality

[We Never Know]

@Polymath257
What exactly do they mean by imaginary numbers? Thanks

"Imaginary numbers might seem like unicorns and goblins — interesting but irrelevant to reality.

But for describing matter at its roots, imaginary numbers turn out to be essential. They seem to be woven into the fabric of quantum mechanics, the math describing the realm of molecules, atoms and subatomic particles. A theory obeying the rules of quantum physics needs imaginary numbers to describe the real world, two new experiments suggest.

Imaginary numbers result from taking the square root of a negative number. They often pop up in equations as a mathematical tool to make calculations easier. But everything we can actually measure about the world is described by real numbers, the normal, nonimaginary figures we’re used to (SN: 5/8/18). That’s true in quantum physics too. Although imaginary numbers appear in the inner workings of the theory, all possible measurements generate real numbers.

Quantum theory’s prominent use of complex numbers — sums of imaginary and real numbers — was disconcerting to its founders, including physicist Erwin Schrödinger. “From the early days of quantum theory, complex numbers were treated more as a mathematical convenience than a fundamental building block,” says physicist Jingyun Fan of the Southern University of Science and Technology in Shenzhen, China.

Some physicists have attempted to build quantum theory using real numbers only, avoiding the imaginary realm with versions called “real quantum mechanics.” But without an experimental test of such theories, the question remained whether imaginary numbers were truly necessary in quantum physics, or just a useful computational tool.

A type of experiment known as a Bell test resolved a different quantum quandary, proving that quantum mechanics really requires strange quantum linkages between particles called entanglement (SN: 8/28/15). “We started thinking about whether an experiment of this sort could also refute real quantum mechanics,” says theoretical physicist Miguel Navascués of the Institute for Quantum Optics and Quantum Information Vienna. He and colleagues laid out a plan for an experiment in a paper posted online at arXiv.org in January 2021 and published December 15 in Nature.

More here
https://www.sciencenews.org/article/quantum-physics-imaginary-numbers-math-reality

 

[Debater Slayer]

An imaginary number is any number consisting of a real number (even if just 1) multiplied by the imaginary unit denoted by i, where i is the square root of -1 and therefore i^2 = -1 (hence the label "imaginary," since negative numbers have no real roots).

Complex numbers--which include a real part and an imaginary part, in the form a + bi--have extensive applications in electric circuits, not just in quantum physics. It has long fascinated me that such counterintuitive mathematical concepts are so useful in many practical applications.

 

[Debater Slayer]

To put my previous response differently, let's think of what the square root Y of any number X is: it's another number that gives you X when you multiply it by itself. So, for example, we know that the square root of 4 is 2 because multiplying 2 by itself gives us 4.

Now, try to do this with a negative number: what number exists that can give you a negative result when you multiply it by itself? A negative number multiplied by itself would cancel out the negative sign and give you a positive result, and you also can never get a negative result from multiplying a positive number by itself. This means that when speaking of real numbers--quantities that can exist in reality--a negative number has no square root: there is no negative number, say -X, for which we can find a real root Y such that Y*Y (i.e., Y^2) equals -X. The negative sign can't result from multiplying any number Y by itself whether Y is negative or positive.

This is where imaginary numbers come in. An imaginary number with a coefficient of 1 is defined as i such that i multiplied by itself--that is, i^2--equals -1. Because such a number doesn't exist in reality, we "imagine" it to exist and define it as the square root of -1.

What follows from this is that any negative number can now have a square root--albeit not a real one. This means that, for example, we can now say that within the domain of the set of complex numbers (as opposed to the set of real numbers, which is a subset of complex numbers), the square root of -25 is 5i, where i^2 gives us the negative sign and 5^2 gives us 25.

I hope this helps, in case my previous answer wasn't detailed enough!

 

[Revoltingest]

While I don't remember any specific applications,
I recall that imaginary numbers were useful in
mechanical engineering.
Remember....math isn't the real world...it's a model
of the real world. One way to see it is that even
counting numbers don't exist. (They're just more
intuitively obvious in application to reality.) People
once thought negative numbers don't exist (only
a few hundred years ago.) Imaginary numbers
are just another kind of thing that doesn't exist.

 

[Heyo]

@Polymath257
What exactly do they mean by imaginary numbers? Thanks

"Imaginary numbers might seem like unicorns and goblins — interesting but irrelevant to reality.

I'm not the Prof, but I have understood maths enough to explain imaginary numbers.

Here is a video about the history of this unusual construct:

As you see, imaginary numbers are a result from ordinary algebra. not something conjured up out of thin air. The moment you have "invented" taking square roots, you are going to encounter cases where you have to take the square root of negative numbers.
(I guess you already know that an imaginary number is the square root of a negative number?)

 

[ChristineM]

not just in quantum physics. It has long fascinated me that such counterintuitive mathematical concepts are so useful in many practical applications.

A non science use for imaginary numbers is 3D graphics where the z dimension is imaginary.

 

[rational experiences]

O stone earth.

Heavens gases.

Theist human conscious is self advice in gas water heavens theories.

Gods stone to convert into his gas heavens wisdom.

Then looks into the cosmos.

Space zero empty. I will say it takes mass closest to a burning gas projected by his rocket thesis to travel by a count say billion years.

In mass a billion he says is straight energy.

In space zero empty is no number. Fake count.

So then he preached gases to travel cooling by billions towards us the energy type would equal stone by space measure cooling and pressure.

Nope wrong. Earth owned a huge pressurized non equated mass first.

No numbers is fakery.

Man says I always tell myself advice as the adult yet I never listen to the adult advice. Meaning an adult human invented science.

His own one self is always newly indoctrinated. Self awareness science told science that science is wrong. First observation is only correct. Natural.

 

[Mock Turtle]

Long forgotten as to details but still remembered vaguely, these form quite a role in the fields of Control Theory and Fluid Dynamics especially during many Mechanical Engineering courses - and hence where a good memory was important to have for exams since in my day we had to remember all such (often complex) formulae rather than have them at hand. Laplace transforms were another little nugget we had to learn too.

Complex Numbers and their Applications
Laplace transform - Wikipedia

 

[LegionOnomaMoi]

@Polymath257
What exactly do they mean by imaginary numbers?

This is a presentation on this work by one of the team members I've made available:

 

[Brickjectivity]

What exactly do they mean by imaginary numbers? Thanks

How it works is that i = sqrt( -1 ) is not technically a number but an amount of rotation. When you multiply by it you are saying "Rotate this around the center either by a fraction of 90 degrees or a number times 90 degrees or both)."

If you look at an imaginary number such as 5 + 3j it is not the same as a cartesian coordinate in (x,y) but is graphed the same way. You go 5 to the right and 3 up, make a dot. That's how you graph 5 + 3j.

In the number 5 + 3i there is a trick. If you multiply this by i you rotate it 90 degrees, and if you mult it by a fraction of i you will rotate it a different number of degrees. For example if you multiply (5 + 3i) times 1/2 of i you will rotate that dot by 45 degrees. Its like you are spinning the dot around where the axis cross.

That is why imaginary numbers are useful. The imaginary plain can do everything that the Cartesian Coordinate system can, plus it has this rotational trick.

i is not actually a number, however it can be symbolically treated as if it is an amount of something. It holds a place. It stands in the way of some operations, and it facilitates others. It is an operator in the equation. It stands in the way because you cannot calculate it, add to it or subtract from it.. It facilitates, because you can pretend to multiple by it, divide by it.

Treat i like it is the square root of negative one, even though that is impossible. If it were a number, you couldn't do that. The fact that its not really a number is a virtue which is carried through to the end of every calculation. i is never resolved, never solved, never known. It is a hinge only. Does not compute, and by not computing it you play the game right.

 

[Shaul]

An imaginary number is any number consisting of a real number (even if just 1) multiplied by the imaginary unit denoted by i, where i is the square root of -1 and therefore i^2 = -1 (hence the label "imaginary," since negative numbers have no real roots).

Complex numbers--which include a real part and an imaginary part, in the form a + bi--have extensive applications in electric circuits, not just in quantum physics. It has long fascinated me that such counterintuitive mathematical concepts are so useful in many practical applications.

Some notations for imaginary numbers use j, not i.

 

[Heyo]

i = sqrt( -1 ) is not technically a number

It is, at least as much as -1 or 1/2 are numbers.
When you start with numbers you usually think about the natural numbers. You can count them and you can add them and all is fine and well.
Then you find that you can also subtract them. Which sound natural at first. You put 5 apples into your basket, you can add another 3 to get eight, you can subtract 6 from them to get 2. Try to take another 6 from them and you just left the set of natural numbers. But that doesn't break mathematics. Just invent/discover the whole numbers and you are fine.
The same is with complex numbers. They are an extension to the set of numbers, no more special than negative or irrational numbers.

 

[Brickjectivity]

It is, at least as much as -1 or 1/2 are numbers.
When you start with numbers you usually think about the natural numbers. You can count them and you can add them and all is fine and well.
Then you find that you can also subtract them. Which sound natural at first. You put 5 apples into your basket, you can add another 3 to get eight, you can subtract 6 from them to get 2. Try to take another 6 from them and you just left the set of natural numbers. But that doesn't break mathematics. Just invent/discover the whole numbers and you are fine.
The same is with complex numbers. They are an extension to the set of numbers, no more special than negative or irrational numbers.

I agree, I think. Since i is not a natural number you cannot have i apples. You can move apples as if they are a point in the imaginary plain only because every point in the imaginary plain corresponds to a point in the real world. You can say that North is the i axis and move your apples 5 i in that direction. You cannot count i apples or 5 i apples.

 

[metis]

Our oldest granddaughter, who's a graduate student in quantum chemistry at the University of Michigan gave a presentation on this math two weeks ago, plus Scientific American had an article on this in an addition a few months ago that I forwarded to her. It was over my head but not hers.