Momentum Eigenstates in 1D Infinite Square Well

[Andy_ToK]

Hi,
I have a question about the momentum eigenstates in a 1D infinite square well example. First of all, are there any eigenstates at all in this example?
By explicitly applying the wavefunction(stationary states) which can be easily obtained from the boundary conditions, it can shown that the energy eigenstate doesn't have a corresponding momentum eigenstate(in the free particle case, each energy eigenstate corresponds to a momentum eigenstate)
I believe it's because the particle is confined that it has a large uncertainty in momentum so no momentum eigenstates exist. But I,myself is not very convinced by this argument.
Any input is appreciated.

 

[Avodyne]

Well, if you consider a finite-potential square well, then the momentum eigenstates still exist in exactly the same sense that they do for a free particle; they just no longer correspond to the energy eigenstates.

For the infinite well, you could take this to mean that the Hilbert space consists of functions on the interval where the potential is zero. Then there are no momentum eigenstates.

(And I'm being mathematically sloppy; mathematicians would say that momentum eigenstates don't exist even for the free particle because they're not normalizable.)

 

[Andy_ToK]

Thanks, Avodyne, but I'm not sure what u mean by "the Hilbert space consists of functions on the interval where the potential is zero". Could you elaborate a little bit?

 

[Avodyne]

If the potential is infinite for (say) |x|>a, then the particle is never allowed to be there; any allowed wavefunction, at any time, must be zero for |x| greater than or equal to a. So, we could take our Hilbert space to be square-integrable functions that vanish for |x| greater than or equal to a. In this Hilbert space, there are no eigenstates of the momentum operator.