In our previous blog post we asked if the Schrödinger equation is Galilean covariant. That is, if the equation has a solution in a frame, can we – in a beautiful way – transform the solution and obtain the solution in a moving frame?
Newtonian mechanics is Galilean covariant. If we have a history of mechanical system, we can – in a simple way – transform the solution to a frame moving at a constant velocity v. The transformed solution satisfies newtonian mechanics in the moving frame.
Huygens's principle
Let us have a wave. Using Huygens's principle, we can easily construct an approximation for the diffraction pattern created by a pinhole in a screen.
|
| | | | ) ) )
|
wave --> v screen diffracted wave
^ y
|
------> x moving frame
--> v
Let the velocity of the wave be v. Let us switch to a moving frame which comoves with the incoming wave.
In the comoving frame, the incoming wave is static. Can we use Huygens's principle for a static wave?
The screen is moving against a static wave. How can the screen create the diffraction pattern?
If we are looking at water waves, then the frame where water does not flow horizontally, can be defined as a preferred frame.
Does it make sense to demand that Huygens's principle should work in any other frame than the preferred frame?
Richard Feynman derived the Schrödinger equation from a path integral, i.e., Huygens's principle
David Derbes (1996) describes how Richard Feynman used a path integral approach to derive the Schrödinger equation. The path integral has much the same idea as in Huygens's principle. If we know the wave function
ψ(t, r)
at a time t₀, we can construct the wave at a later time t₁ by summing the contributions of ψ(t₀, r) for each point r in space.
Huygens said that each point r acts as a new "source" of a new mini-wave. At a later time, the wave's crests are where there is a constructive interference of the mini-waves.
*** WORK IN PROGRESS ***
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