Piece-wise Potential Function

1. The problem statement, all variables and given/known data
Consider reflection from a step potential of height v-knot with E> v-knot, but now with an infinitely high wall added at a distance a from the step.

infinity < x < 0 => v(x) = 0
0≤ x≤ a => v = vknot
x=a => v= infinity

Solve the Schroedinger equation to find ψ(x) for x< 0 and 0 ≤ x ≤ a, solution should contain only one unknown constant.

2. Relevant equations

T-Independent Schrodinger EQ

General forms of a wave function.

3. The attempt at a solution

Is it correct to first assume that all constants are physically possible in both equations? You’ll have a reflection and transmission at the first finite barrier, and a reflection (always) at the infinite barrier. That means there are four constants in both equations. If not, can you explain why?

I should have

ψ_1 = A1 cos (k1*x) + B1 sin(k1*x) (or the respective complex exponentials)
ψ_2= A2 cos(k2*x) + B2 sin(k2*x) (this is for the region 0≤x≤a

When I look at the three boundary conditions,

1. ψ_1(0_ = ψ_2(0)

2. dψ_1/dx (0) = dψ_2/dx (0)

3. ψ_2(a) = 0

I get a complicated algebraic relation between the constants that does not simplify.

So I assume I must get rid of one of the constants, but I’m unsure which one.


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