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.