1. The problem statement, all variables and given/known data Consider the semi-i

Question

1. The problem statement, all variables and given/known data
Consider the semi-infinite square well given by V(x) = -V0 < 0 for 0? x ? a and V(x) = 0 for x > a. There is an infinite barrier at x = 0 (hence the name "semi-infinite"). A particle with mass m is in a bound state in this potential with energy E ? 0. Solve the Schrodinger equation to derive ?(x) for x ? 0. Use the appropriate boundary conditions and normalize the wave function so that the final answer does not contain any arbitrary constants.


2. Relevant equations
[-h_bar2/2m]?'' + V(x)? = E?


3. The attempt at a solution

Now I have 3 equations for 3 unknowns, A1, B1, and B2. But I have been trying to solve this algebraically for quite awhile, and I just can't get it to work. When I solve A1 and B1 in terms of B2 and try to plug them into the third condition, I just get B2 cancelling on both sides. Maybe I'm being really dumb about basic math but I would really appreciate if someone could help with this.

Explanation / Answer

from 1st codition A1 = -B1

3rd condition can be rewritten as A1eik1a - B1e-ik1a = -k2/k1B2e-ik2a

2nd condition is A1eik1a + B1e-ik1a = B2e-ik2a

now add these two equations

2A1eik1a = B2e-ik2a(1-k2/k1)

subtrcat these two equations

2B1e-ik1a = B2e-ik2a (1+k2/k1) put in this B1=-A1 anf compare with above equation

-2A1e-ik1a = B2e-ik2a (1+k2/k1)

but 2A1eik1a = B2e-ik2a(1-k2/k1)

hence

-(1+k2/k1) = (1-k2/k1)

-1 = 1 not possible

hence

there is no consistent solution..

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