Quantum Tunelling Problem with Boundary Conditions

AI Thread Summary
The discussion revolves around solving a quantum tunneling problem with boundary conditions, where the user struggles with setting the correct boundary conditions and understanding the relationships between wavefunctions in different regions. The user initially misinterprets the roles of coefficients A, B, C, and D in the wavefunctions, leading to confusion about the reflection and transmission aspects of the problem. Clarifications are provided regarding the nature of the wavefunctions, particularly that the exponential terms do not blow up as x approaches infinity and the correct interpretation of the coefficients in relation to the potential energy. The conversation emphasizes the importance of correctly applying boundary conditions at specified points to derive meaningful relationships among the unknowns in the wavefunctions. Overall, the thread highlights the complexities of quantum mechanics and the necessity of precise mathematical formulation in solving such problems.
fissile_uranium
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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!

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fissile_uranium said:
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.
 
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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.
 
fissile_uranium said:
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?

fissile_uranium said:
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.

fissile_uranium said:
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##?
 
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