Assume that you have a one dimension box with infinite energy outside, and zero energy from 0 to L. Then my understanding of the Schrodinger equation is that the equation inside will be:(adsbygoogle = window.adsbygoogle || []).push({});

-h^2/2m*d^{2}/dx^{2}ψ = ihd/dtψ

And the energy eigenstates are given by

ψ(x,t) = e^{-iwt}*sin(kx)

where k = n*π/L and w = k^{2}h/2m and n = 1,2,3...

Well, that's nice but what about if there is a positive potential energy U_{0}inside?

-h^2/2m*d^{2}/dx^{2}ψ + U_{0}ψ = ihd/dtψ

What is the solution ψ(x,t) in explicit form? And how do you find the energy eigenstates?

My guess at solving this would be to solve -h^2/2m (ψ)'' = (E-U)ψ as an ODE.

I still get ψ = sin(kx), but I'm not sure whether this satisfies the boundary conditions or not. Also, I get that k^{2}= 2m(E-U)/h^{2}.

My naive assumption is that ψ(0,t) = ψ(L,t) = 0 since it can't exist pass an infinite potential wall and because then the function would be continuous, since ψ = 0 at all other points.

If this is true, then energy eigenstates of the particle must be ψ(x) = Asin(nπ/L*x), where A is any arbitrary phase factor and n = 1,2,3...

Thus, assuming ψ(x,t) = e^{-iwt}ψ(x)

Thus we get h^{2}k^{2}/2m*ψ(x,t)+U*ψ(x,t) = hw*ψ(x,t)

Factoring out, we get h^{2}k^{2}/2m + U = hw = E

Thus w = hk^{2}/2m + U/h = h/2m*(nπ/L)^{2}+ U/h

My interpretation of this is that given a certain n, the eigenstates "look" the same in complex phase space, but they are spinning around faster at a rate of U/h.

Energy is quantized in this box to E = h/2m*(nπ/L)^{2}+ U where n = 1,2,3...

My main problem arises when U is negative, and chosen such that U = -h/2m*(π/L)^{2}, thus leaving the total energy at the ground state as zero, and the wave function is "standing still" with w = 0. What's going on here? Even worse is when U is greater than E_{0}, the angular frequency is negative, does that mean the wavefunction is spinning in the opposite direction as normal positive energy wavefunctions?

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# Infinite square well with finite potential energy inside

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