(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the semi-infinite square well given byV(x)= -V_{0}< 0 for 0≤x≤aandV(x)= 0 forx>a. There is an infinite barrier atx= 0 (hence the name "semi-infinite"). A particle with massmis in a bound state in this potential with energyE≤ 0. Solve the Schrodinger equation to deriveψ(x)forx≥ 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_bar^{2}/2m]ψ'' +V(x)ψ = Eψ

3. The attempt at a solution

- Schrodinger Equation for 0 ≤
x≤aandx>a:

[-h_bar^{2}/2m]ψ'' - V_{0}ψ = Eψ, 0 ≤x≤a

[-h_bar^{2}/2m]ψ'' = Eψ,x≥a- Rewrite Schrodinger equations:

ψ'' + 2m(E+V_{0})/h_bar^{2}= 0, 0 ≤x≤a

ψ'' + 2mE/h_bar^{2}= 0,x≥a- Solve Schrodinger equations:

ψ_{1}= A_{1}e^{ik1x}+ B_{1}e^{-ik1x}, k_{1}= sqrt[2m(E+V_{0})]/h_bar, 0 ≤x≤a

ψ_{2}= A_{2}e^{ik2x}+ B_{2}e^{-ik2x}, k_{2}= sqrt[2mE]/h_bar,x≥a- k
_{2}is negative, and the wave function must not blow up atx= ∞, so A_{2}= 0:

ψ_{1}= A_{1}e^{ik1x}+ B_{1}e^{-ik1x}, k_{1}= sqrt[2m(E+V_{0})]/h_bar, 0 ≤x≤a

ψ_{2}= B_{2}e^{-ik2x}, k_{2}= sqrt[2mE]/h_bar,x≥a- Apply boundary conditions:

ψ_{1}(0) = 0

ψ_{1}(a) = ψ_{2}(a)

ψ'_{1}(a) = ψ'_{2}(a)

1st condition: A_{1}+ B_{1}= 0

2nd condition: A_{1}e^{ik1a}+ B_{1}e^{-ik1a}= B_{2}e^{-ik2a}

3rd condition: ik_{1}A_{1}e^{ik1a}- ik_{1}B_{1}e^{-ik1a}= -ik_{2}B_{2}e^{-ik2a}

Now I have 3 equations for 3 unknowns, A_{1}, B_{1}, and B_{2}. But I have been trying to solve this algebraically for quite awhile, and I just can't get it to work. When I solve A_{1}and B_{1}in terms of B_{2}and try to plug them into the third condition, I just get B_{2}cancelling on both sides. Maybe I'm being really dumb about basic math but I would really appreciate if someone could help with this. Thanks!

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# Homework Help: Find the wave function of a particle bound in a semi-infinite square well

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