Question regarding 1-D PIW thought problem

[karma345]

We know that all states of the wavefunctions must be quantized. Therefore, when we have a particle, say an electron, trapped in a well with infinite potentials on either side - let's set the boundaries to the traditional -1/2L to 1/2L - the ground state of the energy must give us a wavelength which must be, at most, 1/2lambda=L. We then can have n number of 1/2lambdas within the well, and we can describe those states alternatingly with Cos and Sin functions - but n must be a whole integer - otherwise the state is not quantized. Now, suppose we inject a single electron, at a velocity of .01c, let's be more specific about it and say 1x10^6 m/s, into a well that is 4 Angstroms wide - i.e. from -1/2L to 1/2L, we have a space spanning 4 Angstroms. We know, from the DeBroglie relation that the wavelength of the electron at that speed is approximately 7 Angstroms. What happens when the electron enters that well? Will the energy state immediately adapt to a quantized level - and if so, will it immediately fall to ground state or will it adapt to the nearest frequency which allows whole numbers of half wavelengths in the 4 Angstrom well? Or does the whole system just break down?

Jason

 

[vanesch]

Hello,

What you describe are the so-called stationary states, that means, states with a definite energy value (which also have the property of not evolving in time - up to a phase factor). But in quantum mechanics, there is the fundamental principle of superposition, so many more states than these stationary states are allowed. When you have "initial conditions" as you describe them which do not correspond to a stationary state, you will in fact have to describe it as a superposition of stationary states : in the case you cite, this would come down to writing down a Fourier series in which you devellop the initial state.
So you simply have a non-stationary state.

cheers,
Patrick.

 

[karma345]

Hello,

When you have "initial conditions" as you describe them which do not correspond to a stationary state, you will in fact have to describe it as a superposition of stationary states : in the case you cite, this would come down to writing down a Fourier series in which you devellop the initial state.

cheers,
Patrick.

I see. That makes perfect sense. The text never reallly makes that abundantly clear, althought I can see where it is implied in the context. Thanks.

Jason