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deedsy
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Homework Statement
The potential for a particle mass m moving in one dimension is:
V(x) = infinity for x < 0
= 0 for 0< x <L
= V for L< x <2L
= infinity for x > 2L
Assume the energy of the particle is in the range 0 < E < V
Find the energy eigenfunctions and the equation that determines the energy eigenvalues. Don't worry about normalizing the eigenfunctions.
Homework Equations
The Attempt at a Solution
I believe I've found the equation for the energy eigenvalues...
in the region 0 < x < L the wave equation has the form:
[itex] \psi_1(x) = e^{ikx} + Be^{-ikx} [/itex] where [itex] k = \sqrt{2mE}/\hbar[/itex]
for L < x < 2L, the wave equation is:
[itex] \psi_2(x) = Ce^{-qx} + De^{qx} [/itex] where [itex] q = \sqrt{2m(E-V)}/\hbar[/itex]
applying the boundary condition [itex] \psi_1(x=0) = 0 [/itex],
I find B = -1 and the wave equation can be rewritten and simplified as [itex] \psi_1(x) = A' sin(kx) [/itex]
applying the boundary condition [itex] \psi_2(x=2L) = 0 [/itex],
I find [itex]D = -C e^{-4qL} [/itex] and the wave equation can be simplified to [itex] \psi_2(x) = C' sinh[q(x-2L)] [/itex]
Now, equating these two equations at x = 0 and their derivatives at x=L, and then dividing the two results gave me a solution i can use to solve for E, the energy eigenstates: [itex] q tan(kL) = -k tanh(qL) [/itex]
But now, I'm not sure how to go about finding the energy eigenfunctions... For the plain infinite well, it was easy because you just had one equation, sin(something) = 0 , so something = n*pi. Then you just rewrote the equation with n in it...
I'm thinking this involves another application of the boundary conditions, but I don't see how quantized energy functions will arise from equating my solutions at x = L...