Quantum Tunelling Problem with Boundary Conditions

[fissile_uranium]

Homework Statement

a) Assume a particle is sent from x = +∞ towards x = 0 with amplitude A and it is reflected to a
right moving wave with amplitude B. Find a relation between A and B in terms of E, V0, and a (and possibly some fundemental constants.)

b) Calculate the norm |B/A| and explain what your result implies? Is it physically sensible? (Hint:
for a complex number (u + iv)/(a + ib) with real numbers u, v, a, b, find the norm.)

c) Let −e^i2δ be the phase of the ratio B/A and find an expression for the so-called phase shift δ.

Relevant Equations

Shrödinger's Equation

Questions and the figure is given. This is a task from previous years, fortunately there are no answers.

I have tried to solve, starting from part a, but I think I do not understand how to set boundary conditions. I ended up with:

2*C*e^(i*a*k2)=B*e^-i*a*k1*(1-(i*k1/i*k2)

When I proceed with part b, I get (k2-k1)^2/(2*k2)^2

And this does not seem coincide wih the Hint.

From beginning.
I set my equations as:
For Region 2 (0-pot):

Ae^ik1x + Be^-ik1x

and For Region 3:
Ce^ik2x + De^-ik2x

Later on I eliminated wave part with coefficient A since it blows up as x goes to inf. And using ψ(x) and ψ'(x) I eliminated the part with coeff D. So I ended up with a relation between B and C term, two unknowns (which is meaningful). But I think I interpreted the reflection wrong. Should it be a relation between A and B terms?

I also thougt, since the wall has inf. pot. can I take it as a full reflection and amplitude 1=B=C? and move on with finding A and B? It also seems like there would be a on-going reflection loop in the well, is it important?

I appreciate any help and insightful comment! I am new with modern physics, every recommendation is appreciated.

Thanks!

 

[TSny]

From beginning.
I set my equations as:
For Region 2 (0-pot):

Ae^ik1x + Be^-ik1x

and For Region 3:
Ce^ik2x + De^-ik2x

Later on I eliminated wave part with coefficient A since it blows up as x goes to inf.

Labeling the two regions as "Region 2" and "Region 3" and using $k_1$ for Region 2 and $k_2$ for Region 3 could be confusing.

How are $k_1$ and $k_2$ related to $E$ and $V_0$?

It looks like you used the coefficient $A$ for a part of the wavefunction in Region 2: $0 \leq x \leq a$. But the statement of the problem denotes $A$ as the coefficient for the part of the wavefunction coming in from infinity. So, the part of the wavefunction with coefficient $A$ should be for the region $a \leq x < \infty$, (Region 3).

You said you eliminated $Ae^{ik_1 x}$ since this part blows up as $x$ goes to $\infty$.
If $k$ is a real number, $e^{ikx}$ does not blow up for $x \rightarrow \infty$. $e^{ikx}$ always has magnitude equal to $1$ for real $k$.

For a particle with wavenumber $k$ moving to the left, would you use $e^{ik x}$ or $e^{-ik x}$? (Assume $k$ is a positive real number.)

EDIT: Since $E > V_0$, there is no "tunneling" in this problem.

 

[fissile_uranium]

You are correct, I must have said Scattering Problem.

I have written $k_2= \sqrt{(2m*E/ \hbar^2)}$ and $k_1= \sqrt{(2m*(E-u_0)/ \hbar^2)}$

You are also correct about that part not blowing up but I should not solely take exponential parts as 1 right? Now I end up with 4 unknowns, only being able to eliminate one wave equation from derivative. How can I find a relation between two items?. Also I am taking $e^{(-ikx)}$ as left moving wave.

Can you help me about boundary conditions? Also, sorry for late responding, thanks.

 

[TSny]

You are correct, I must have said Scattering Problem.

I have written $k_2= \sqrt{(2m*E/ \hbar^2)}$ and $k_1= \sqrt{(2m*(E-u_0)/ \hbar^2)}$

OK. Here $u_0 = V_0$.
Edit: Looks like you have the subscripts "1" and "2" switched. Do you agree?

You are also correct about that part not blowing up but I should not solely take exponential parts as 1 right?

I'm not sure what "1" refers to here. In the region $0 < x < a$ you can use a superposition of exponentials $\psi_1(x) = C e^{ik_1 x} + D e^{-ik_1 x}$ or, equivalently, you can use $\psi_1(x) = C \sin(k_1 x) + D \cos(k_1 x)$. If you think about the boundary condition at $x = 0$, you should be able to see which of these choices is better for this problem.

Now I end up with 4 unknowns,

Yes. For the region $x > a$ we have $\psi_2(x) = C_1 e^{ik_2 x} + C_2 e^{-ik_2 x}$. Here, I've used the notation $C_1$ and $C_2$ for the constants. But, one of these constants is $A$ and the other is $B$, where $A$ and $B$ are as given in the problem statement. Which of $C_1$ and $C_2$ is equal to $A$ and which is $B$?

Can you state the boundary conditions that must be satisfied at $x = 0$ and $x = a$?