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Quantum physics falls apart without imaginary numbers

scientificamerican.com

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Re: Quantum physics falls apart without imaginary numbers

#41

Post by digging »

fullstackchris wrote: post #30
andrew_eu wrote: post #15

There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term "imaginary numbers" because it implies some mystery around them. I vaguely remember reading that he preferred the term "lateral" numbers, but that may be a mistake. Euler's formula connects them very plainly with rotations in a complex number plane. The intuition I developed with the…

The best way I've ever had it explained to me is with electron tunneling. You ask, how did the electron "jump" that potential hill, when it actually didn't have the momentum to do so? The answer: it didn't, it quite literally "went through" the potential hill. So you ask, well, what kind of momentum (mv^2) would allow for this "tunneling" momentum? You invariably arrive at a _negative_ moment... and thus only an imag…

I thought quantum tunneling was due to uncertainty, allowing the position of a particle to resolve on the other side of a barrier sometimes because it is undefined before the tunneling.

Are we talking about different things? Is one of us way off the mark? Or are these two angles of the same phenomenon?

Re: Quantum physics falls apart without imaginary numbers

#42

Post by lanza »

fullstackchris wrote: post #30
andrew_eu wrote: post #15

There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term "imaginary numbers" because it implies some mystery around them. I vaguely remember reading that he preferred the term "lateral" numbers, but that may be a mistake. Euler's formula connects them very plainly with rotations in a complex number plane. The intuition I developed with the…

The best way I've ever had it explained to me is with electron tunneling. You ask, how did the electron "jump" that potential hill, when it actually didn't have the momentum to do so? The answer: it didn't, it quite literally "went through" the potential hill. So you ask, well, what kind of momentum (mv^2) would allow for this "tunneling" momentum? You invariably arrive at a _negative_ moment... and thus only an imag…

Classical mechanics are fundamentally wrong. They are low energy approximations to reality. "How did the bal go through the hill when it actually didn't have the momentum to do so" is a nonsense question because the equations of motion you are attempting to use to describe the phenomena are wrong.

You can't and shouldn't try to understand QM from a CM standpoint. If you remember Taylor series expansions, this is like trying to understand `sin(x)` by looking at it's first order Taylor series expansion `x`. Your question about momentum and a potential hill is the same question as "how did the value of sin(x) start decreasing if `x` is linear?" You're using too few terms of the series expansion. The mechanisms of Newton's laws are first order terms of a proper QM solution.

Re: Quantum physics falls apart without imaginary numbers

#44

Post by [deleted] »

nathan_compton wrote: post #16
tehsauce wrote: post #13

“Complex numbers” are rather poorly named. They are more naturally understood as simply a vector which has a magnitude, can be rotated and scaled. As geometric objects they are much more intuitive. The subject geometry algebra takes a great approach of generalizing this idea and augmenting basic linear algebra to unify complex numbers and beyond (quaternions, ect) with geometric objects and operations. This also fits…

You miss a key part of complex numbers if you think of them as just vectors: they are a field.

[deleted]

Re: Quantum physics falls apart without imaginary numbers

#45

Post by rcme »

lanstin wrote: post #27
tehsauce wrote: post #13

“Complex numbers” are rather poorly named. They are more naturally understood as simply a vector which has a magnitude, can be rotated and scaled. As geometric objects they are much more intuitive. The subject geometry algebra takes a great approach of generalizing this idea and augmenting basic linear algebra to unify complex numbers and beyond (quaternions, ect) with geometric objects and operations. This also fits…

Complex analysis is so much more regular than real analysis differentiability over a two dimensional quantity is so much strong than over a one dimensional quantity that you have much stronger results. Basically, if you know an analytic function in a neighborhood you know it over the entire plane. Plus you have functions like e ^ ( 1 / z ) which is pretty amazing around zero.

While what you say is true, I could never intuitively grasp that properties of analytic functions. Like I could read and understand the proofs, as in follow one step to the next, but I could never succinctly describe, intuitively, why one should expect the proofs to hold. Even the most fundamental concepts in complex analysis are more like facts rather than logical deductions (to me).

Re: Quantum physics falls apart without imaginary numbers

#47

Post by BrandoElFollito »

nathan_compton wrote: post #17
BrandoElFollito wrote: post #8

Earlier quoted context omitted.

This is one of the complicated steps in physics (I have a PhD in physics (and forgot everything since)). First you have some math that goes along discovering physics. You split vectors, multiply mass by something and it's fine. Then you have math that helps you with physics. Simple differentials equations that uncover while laws of nature (cooling down speed for instance). This is the golden time for many because you…

There really isn't anything weird or suspect about renormalization (except the name, perhaps). Read A. Zee's book on Quantum Field Theory.

You are certainly right, like I said it was a long time ago.

But doing splits and advanced acrobatics to get rid of infinities always felt like a hack (Feynman felt the same do at least I am not alone :))

Re: Quantum physics falls apart without imaginary numbers

#48

Post by bmacho »

retrocryptid wrote: post #3

One of the first things we were taught in physics was "don't think that imaginary or complex numbers have physical significance. just do the math." And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out.

Same holds for negative numbers. There are no negative quantities in physics, negative numbers as quantities only appear if you order your equations wrong. (And one can argue against the other appearances of negative numbers and minus signs.)

Re: Quantum physics falls apart without imaginary numbers

#49

Post by [deleted] »

sebzim4500 wrote: post #31
whatshisface wrote: post #20

Earlier quoted context omitted.

"Quantum physics falls apart without imaginary numbers." > Marco made a curious face, so Toni posed the question: “Can standard quantum theory work without imaginary numbers?”

By that logic, it doesn't even need real numbers. Just do everything with cauchy sequences of rationals.

[deleted]

Re: Quantum physics falls apart without imaginary numbers

#50

Post by TechnicolorByte »

lanza wrote: post #42
fullstackchris wrote: post #30

Earlier quoted context omitted.

The best way I've ever had it explained to me is with electron tunneling. You ask, how did the electron "jump" that potential hill, when it actually didn't have the momentum to do so? The answer: it didn't, it quite literally "went through" the potential hill. So you ask, well, what kind of momentum (mv^2) would allow for this "tunneling" momentum? You invariably arrive at a _negative_ moment... and thus only an imag…

Classical mechanics are fundamentally wrong. They are low energy approximations to reality. "How did the bal go through the hill when it actually didn't have the momentum to do so" is a nonsense question because the equations of motion you are attempting to use to describe the phenomena are wrong. You can't and shouldn't try to understand QM from a CM standpoint. If you remember Taylor series expansions, this is like…

Incredible analogy! The closer is a keeper:

> The mechanisms of Newton’s laws are first order terms of a proper QM solution.

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