Quantum particle in a rigid box with 2 given wavefunctions solving for energies

I have a quantum problem that I can't seem to figure out:

There's an electron in a 1-D rigid box of length 2A but it is known to reside in a central segment of 1A with uniform probability of residing within this segment.
There are two possible wavefunctions:
one with constant phase: ψ(x)= 1/√a (a=1A)
and with varying phase: ψ(x)= e^(jxpi/4a)/√a
Both of those wave equations are for within the segment and outside the segment ψ(x)= 0.

Determine the 3 lowest energies expected for each case and the probability of each outcome.

Since the wave equation is 0 outside the segment, I thought I'd treat this as a regular particle in a box problem with the box of length 1A. Then I plugged the ψ(x) given into the Schrodinger's equation to find E for the 2 cases:

d^2/dx^2 ψ(x) = -(2m/h^2)*E*ψ(x)

However, for the first case of constant phase, this gives that E=0 and for the second case, E= (h*pi)^2/(32ma^2) which is really just a constant and I can't get the 3 lowest energies.

What am I doing wrong? Any help would be appreciated!

Answers and Replies

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DrDu
Science Advisor
You would only get E=0 if psi were constant everywhere, but at 0.5 A and 1.5 A, the second derivative of the wavefunction is certainly not 0.

My understanding was that from 0-0.5 and 1.5-2, ψ(x)=0 (for both cases), so the second derivative would be 0 as well, no? And within 0.5-1.5 it is just a constant 1/√a so that derivative is also 0.

DrDu
Science Advisor
Nevertheless at these very two points the derivatives are not well defined.