# Constraints on Quantum Wave

I need help conceptually visualizing the mechanics of an EM wave, and 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. "Normal" force waves result from modulations and/or imbalances between competing forces on a system within the constraints of the system in question. For example, as sound waves travel through a medium, the momentum of the particles and the electromagnetic repulsion forces the waves forward into modulating high and low density crests and troughs. Water waves also occur as the particle momentum and EM repulsive forces displace the water as allowed by the competing force of gravity, producing the modulation of the wave. Tensile forces and momentum modulation in a vibrating guitar string are relatively easy to imagine. These descriptions are all obviously over simplifications, but the general ideas are intuitively easy to visualize. But what are the competing/constraining factors that produce the "wave" modulation of light or DeBroglie particle waves? Can anyone help me with an intuitive image here?

## Answers and Replies

mfb
Mentor
The Schrödinger equation.
I don't think there is an appropriate classical picture.

• bhobba
Thanks. But aside from the mathematical construct of the state vector...? Perhaps an intuitive classical image of a De Broglie type of particle wave was too much to ask. If I limit my question to an electromagnetic "wave"... there is no ether, so what is it that is "waving"? What is oscillating, and why?

I know that the wave length is correlated to the energy content of the photon. That's not what I'm going for. I'm simply noting that classical waves are intuitively logical as to what is causing the wave mechanics, and was hoping for a similar idea with regard to EM radiation. It's just a concept I've always struggled with.

I've struggled with it too, Feeble. From our human "perceptual" position, it just "seems" like a wave needs a medium (like water or air). However, as I understand them, there is no medium for EM waves. Furthermore, it's a mistake to think of them as a particle that is vibrating.

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

bhobba
Mentor
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

bhobba
Mentor
Thanks. But aside from the mathematical construct of the state vector...? Perhaps an intuitive classical image of a De Broglie type of particle wave was too much to ask. If I limit my question to an electromagnetic "wave"... there is no ether, so what is it that is "waving"? What is oscillating, and why?

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

jfizzix
Science Advisor
Gold Member
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,
$\frac{\partial^{2}A}{\partial x^{2}}+\frac{\partial^{2}A}{\partial y^{2}}+\frac{\partial^{2}A}{\partial z^{2}}=-k^{2} A$,
and use the small angle approximation (so as to describe laser beams propagating along the z-axis), you get the paraxial helmholtz equation
$-\frac{1}{2}\big(\frac{\partial^{2}A}{\partial x^{2}} +\frac{\partial^{2}A}{\partial y^{2}}\big) = i k \frac{\partial A}{\partial z}$.

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

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.

Last edited:
• mfb, Greg Bernhardt and bhobba