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Semi-infinite square well

Consider the semi-infinite square well given by V(x)=-Vo<0 for 0<=x<=a and V(x)=0 for x>a. There is an infinite barrier at x=0. A particle with mass m is in a bound state in this potential energy E<=0.

a) solve the Schroedinger Eq to derive phi(x) for x=>0. Use the appropriate boundary conditions and normalized the wave function so that the final answer does not contain any arbitrary constants.

b) Show that the allowed energy levels E must satisfy the equation (attached)

c)The equation in part (b) can not be solved analytically to give the allowed energy levels but simple solutions exist in certain special cases. Determine the conditions on Vo and a so that a bound state exists with E=0.

I started this problem but am stuck because E is both equal to zero and less than zero. I am looking for an outline for the problem.


Solution Preview

The wavefunction is continuous and the derivative is also continuous, except where the potential jumps by an infinite ampunt. So, at x= 0 the wavefunction is continuous, but the derivative need not be continuous there. at x = a, bit the wavefunction and its derivative are continuous.

Since for x<0 the wavefunction must be zero, it follows from continuity at x=0 that phi(0) = 0

The Schrödinger equation in the region 0<x<a is:

-h-bar^2/(2m) psi''(x) = (E + V_0) psi(x)

We can write the solutions of this equation as:

psi(x) = A Sin(k x) + C Cos(k x)


k = sqrt[2m(E+V_0)]/h-bar

Demanding that psi(0) = 0 yields C = 0. So we have:

psi(x) = A Sin(k x) (1)

In the region x>a, the Schrödinger equation ...

Solution Summary

A detailed solution is given.