Solving Schrodinger Equation for Modified Stepwise Potential

[beebopbellopu]

Homework Statement

Consider reflection from a step potential of height V₀ with E > V₀, but now with an infinitely high wall added at a distance a from the step (x=0 at V=V₀)

Solve the Schrodinger equation to find [tex]\varphi[/tex](x) for x<0 and 0 [tex]\leq[/tex] x [tex]\leq[/tex] a. Your soultion should contain only one unknown constant.

Homework Equations

Time independent Schrodinger equations for x<0 and 0 [tex]\leq[/tex] x [tex]\leq[/tex] a regions.

For x<0: [tex]\varphi_{1}[/tex] = A₁e^ik₁x + B₁e^-ik₁x where k₁ = [tex]\frac{\sqrt{2mE}}{\hbar}[/tex]

For 0 [tex]\leq[/tex] x [tex]\leq[/tex] a: [tex]\varphi_{2}[/tex] = A₂e^ik₂x + B₂e^-ik₂x where k₂ = [tex]\frac{\sqrt{2m(E-V₀)}}{\hbar}[/tex]

The Attempt at a Solution

Sorry if my formatting is a little off, this is my first time posting. Anyways.

So I tried enforcing my boundary conditions:

1) [tex]\varphi_{1}[/tex](0) = [tex]\varphi_{2}[/tex](0)
2) [tex]\frac{d\varphi_{1}}{dx}[/tex] evaluated at x=0 is equal to [tex]\frac{d\varphi_{2}}{dx}[/tex] evaluated at x=0
3) [tex]\varphi_{2}[/tex](a) = 0

which resulted in the following (respectively to each boundary condition):

1) A₁ + B₁ = A₂ + B₂
2) k₁(A₁ - B₁) = k₂(A₂ - B₂)
3) A₂e^ik₂a = -B₂e^ik₂a

Then using the 3rd condition, you get A₂ = -B₂e^-2ik₂a

plugging that into condition 1 yields A₁ + B₁ = B₂(1-e^-2ik₂a)

Then from the 2nd condition I get A₁ = B₁ - [tex]\frac{-B_{2}k_{2}}{k_{1}}[/tex](e^-2ik₂a+1) which I then plugged back into the equation from condition 1.

So along with the equation for A₂, I get the following constants:

B₁ = [tex]\frac{B_{2}}{2}[/tex][(1-e^-2ik₂a)+[tex]\frac{k_{2}}{k_{1}}[/tex](1+e^-2ik₂a)]

and A₁ = B₂[[tex]\frac{1}{2}[/tex] - [tex]\frac{k_{2}}{k_{1}}[/tex] - e^-2ik₂a([tex]\frac{1}{2}[/tex] + [tex]\frac{k_{2}}{k_{1}}[/tex])]

Do I then just plug these directly back into the schrodinger equations? My main question is that I feel like there is a way to simplify these constants in a way I'm just not seeing, and hoping if there is a way someone could point me in the right direction. Thanks.

 

[mark726]

Thank you for your question. It seems like you have made good progress in solving the Schrodinger equation for this problem. However, you are correct in thinking that there may be a simpler way to express the constants in your solution.

One way to simplify the constants is to use the fact that the wave function must be continuous at the boundary points. This means that the values of A1 and B1 must be equal to the values of A2 and B2 at x=0 and x=a. This can help you eliminate some of the constants in your solution.

Another approach is to use the boundary conditions to solve for the unknown constant in terms of known quantities. For example, you can use the third boundary condition to solve for B2 in terms of A2. This will then allow you to express A1 and B1 in terms of A2, which will simplify your solution.

I hope this helps. Keep up the good work!