QM binding energy and reflection coefficient

[Felicity]

Homework Statement

consider a potential given by

V(x) = infinity x < 0
= 0 x > a
= a negative function of x in between

suppose it is known that the interior wave function is such that

(1/u) (du/dx) at x=a = f(E)

a. what is the binding energy of a bound state in terms of f(Eb)?

b. Suppose f(E) is a very slowly varying function of E so that we can take it to be a constant. Calculate the reflected amplitude R(k) in terms of f if the wave function for x > a has the form
e^-ikx + Re^ikx, and check that

(the absolute value of R(k))² = 1

2. Relavant equations

k²= 2mE/hbar²

The Attempt at a Solution

a. to find the binding energy I calculated f(E) by taking u at x > a to be Te^ikx so that
1/u du/dx = ik

I know that the bound states are discrete solutions where E < 0 and that in order to find them I must match the u at x > a to u in the well at x = a but how do I do this if I don't know the equation for u in the well?

b. How can the wave function at x > a be e^-ikx + Re^ikx?

why is it important that f(E) be a constant?

where do I start?


Thank you,

Felicity

 

[Jhero]

Dear Felicity,

Thank you for your interesting questions. I will do my best to provide some guidance and assistance in solving them.

a. To find the binding energy, we can use the fact that the wave function must be continuous at x=a. This means that the value of u at x=a for x > a must match the value of u for x < a. We can use this to determine the value of f(E) at x=a. Once we have f(E) at x=a, we can use the fact that the potential is zero for x > a to solve for the binding energy in terms of f(E).

b. The wave function at x > a can be written as e-ikx + Reikx because it satisfies the boundary conditions for a potential well. The first term represents the incident wave, while the second term represents the reflected wave. It is important for f(E) to be a constant because it allows us to simplify the calculation of the reflected amplitude R(k). We can use the fact that f(E) is slowly varying to approximate it as a constant, making the calculation easier.

To start, you can use the boundary conditions and the fact that the potential is zero for x > a to determine the value of f(E) at x=a. Then, you can use this value to solve for the binding energy in terms of f(E). For part b, you can use the fact that f(E) is a constant to simplify the calculation of R(k) and then check that |R(k)|^2 = 1. I hope this helps. Good luck with your calculations! (The Scientist)