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- Thread starter Feeble Wonk
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mfb

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The Schrödinger equation.

I don't think there is an appropriate classical picture.

I don't think there is an appropriate classical picture.

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It helps me to remember that it's best to think of them as transverse waves (like an ocean wave, and not like a sound wave which is is longitudinal (compressive and expansive)). Also, it helps me to break apart the electrical (E) component from the magnetic (B) component. All by itself, the electrical component can be decomposed into an X axis and a Y axis component (assuming that Z is the direction of progression). Focusing just on the E component (and it's X and Y components) allows me to understand linear, elliptical, and circular polarization as phase discrepancies between the X and Y components. And, even when the X and Y components are "in phase", the magnitude of the X component to the Y component gives the angle of the linear polarization.

Also, there are some excellent YouTube videos that illustrate this. However, I'll let you find them.

Best of Luck,

Elroy

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bhobba

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Iand especially a DeBroglie type of quantum "wave". I realize that it's a probability wave, but I'm trying to extrapolate a classical image to the general idea.

There is no such thing as a quantum wave.

What is usually meant is the free particle solution to Schroedinger's equation which is wave-like (see 5.3):

http://www.physics.ox.ac.uk/Users/smithb/website/coursenotes/qi/QILectureNotes3.pdf

It not classical - forget such ideas - they are wrong.

Thanks

Bill

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bhobba

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The EM field.

The reason the EM field exists is so energy and momentum is conserved as required by Noethers theorem. Something is required for it to be stored - that is the field. This is implied by certain no go theorems worked out by Wigner.

Thanks

Bill

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The Schrödinger equation.

I don't think there is an appropriate classical picture.

Remarkably, there kind of is!

If you take the optical Helmholtz equation,

[itex]\frac{\partial^{2}A}{\partial x^{2}}+\frac{\partial^{2}A}{\partial y^{2}}+\frac{\partial^{2}A}{\partial z^{2}}=-k^{2} A [/itex],

and use the small angle approximation (so as to describe laser beams propagating along the z-axis), you get the paraxial helmholtz equation

[itex]-\frac{1}{2}\big(\frac{\partial^{2}A}{\partial x^{2}} +\frac{\partial^{2}A}{\partial y^{2}}\big) = i k \frac{\partial A}{\partial z} [/itex].

This is mathematically identical to the Schrodinger equation (up to constants and such, and switching z for time) for a free particle moving in two dimensions

[itex]-\frac{\hbar^{2}}{2m}\big(\frac{\partial^{2}\Psi}{\partial x^{2}} +\frac{\partial^{2}\Psi}{\partial y^{2}}\big) = i \hbar \frac{\partial \Psi}{\partial t} [/itex]

In short, how a laser beam spreads as a function of propagated distance z, has the same form as a 2D free particle evolving in time. If you add a spatially varying index of refraction, you can have an effective potential energy too, giving you a kind of optical schrodinger equation. It's useful for describing light in fiber optics.

It's maybe not a truly classical analogue, but it's food for thought.

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