Solving Particle Incident on Rectangular Barrier

[hasan_researc]

b]1. Homework Statement [/b]

Consider a particle of mass m incident on a rectangular barrier with a potential

V(x) = V₀ for 0 $$\leq$$ x < w
V(x) = 0 otherwise,

where V₀ is a positive constant. The particle has an energy E > V₀. Find expressions in terms of E and V₀ for the magnitudes of the associated wavenumbers inside and outside the barrier, k_i and k_o, respectively.

Homework Equations

The Attempt at a Solution

E = p^/2m = (h-cross*k)²/2m

Therefore, E = $$\frac{(h-cross*k_{o})^{2}}{2m}$$ gives k$$_{o}$$ and
E - V$$_{o}$$ = $$\frac{(h-cross*k_{i})^{2}}{2m}$$ gives k$$_{i}$$.

Am I right?

 

[fzero]

Looks good to me.

 

[hasan_researc]

Thanks!

Next I have been asked the following:

For a particle incident from the left, write down appropriate general forms of wavefunctions
which satisfy the time-independent Schrodinger equation for each of the
three regions, namely x < 0, 0 $$\leq$$ x < w, and w $$\leq$$ x.

This is my answer:

x < 0 : u(x) = Ae^ik₀x + B^-ik₀x

0 $$\leq$$ x < w : No idea

w $$\leq$$ x : No idea

Thanks in advance for any help.

 

[fzero]

Try using sums of $$e^{\pm ikx}$$ in both locations. Use the Schrodinger equation to relate k in each region to k_o or k_i and boundary conditions to relate the coefficients to A and B.

 

[hasan_researc]

Thanks!

So, to begin, I'll write the solution as follows:

0 $$\leq$$ x < w : u₂ = Ce^ik_ix + De^-ik_ix

w $$\leq$$ x : u₃ = Ee^ik_ix + Fe^-ik_ix.

I don't think any of the coefficients are zero, so I'm guessing the psi-squared is sinusoidal in all three regions.

I don't understand what you have meant when you said "Use the Schrodinger equation to relate k in each region to k_o or k_i ".

I'll express C, D, E and F in terms of A and B later.

Anyway, to continue, the next part of the question goes as follows:

For the particular case where the particle energy is $$E = V_{0} + \frac{(hcross*\pi)^{2}}{2mw^{2}}$$ ,
calculate the wavenumber k$$_{i}$$ , and hence wavelength, of the wavefunction in terms of the barrier width w.

The wavefunction and its derivative must be continuous at x = 0 and x = w. For
a particle with the energy given in part 3 above, use these continuity requirements
to find an expression for the ratio of the transmitted wavefunction amplitude to the
incoming wavefunction amplitude.

Find the value of the barrier transmission coefficient for this energy.

This is how I've answered it.

Firstly, substituting the given expression for E into the formula for k_i gives a value for k_i of pi/w.

Next, I have used the boundary conditions as follows.

u₁ = u₂ at x = 0 : A + B = C + D
Same for derivatives : A - B = $$\frac{k0}{ki}$$ (C - D)

u₂ = u ₃ at x = w: C - D = Ee^ik_ix + Fe^-ik_ix
Same for derivatives : C + D = $$\frac{k0}{ki}$$ Ee^ik_ix + Fe^-ik_ix


I'm not sure what to do next.

PS: I'm sorry if this is along question. There are only three more parts to it.

 

[fzero]

Thanks!

So, to begin, I'll write the solution as follows:

0 $$\leq$$ x < w : u₂ = Ce^ik_ix + De^-ik_ix

w $$\leq$$ x : u₃ = Ee^ik_ix + Fe^-ik_ix.

I don't think any of the coefficients are zero, so I'm guessing the psi-squared is sinusoidal in all three regions.

I don't understand what you have meant when you said "Use the Schrodinger equation to relate k in each region to k_o or k_i ".

u₂ and u₃ must be solutions to the Schrodinger equation in the appropriate regions. You need to check this and, if necessary, modify them so that they are. Hint: u₃ is not a solution.

As far as whether or not all coefficients are nonzero, you need to consider the physical situation of this being a problem about an incident particle on the barrier. After you decide whether the incident particle is right or left moving, you will need to decide which boundaries allow reflection and whether you will have right and left moving components in every region.

Next, I have used the boundary conditions as follows.

u₁ = u₂ at x = 0 : A + B = C + D
Same for derivatives : A - B = $$\frac{k0}{ki}$$ (C - D)

The 2nd equation is wrong, you should recheck your algebra and get the ratio of wavenumbers correctly. When you've got the correct equations you can solve for C and D in terms of A and B.

u₂ = u ₃ at x = w: C - D = Ee^ik_ix + Fe^-ik_ix
Same for derivatives : C + D = $$\frac{k0}{ki}$$ Ee^ik_ix + Fe^-ik_ix

Since u₃ is not a correct solution, you'll need to redo these when you get that sorted out. In your first equation you haven't evaluated the RHS at x = w. When you sort the solution and algebra out, you will be able to solve for E and F in terms of C and D, however you still need to sort out whether all of the A, B,...F are nonzero on physical grounds.

 

[hasan_researc]

u₂ and u₃ must be solutions to the Schrodinger equation in the appropriate regions. You need to check this and, if necessary, modify them so that they are. Hint: u₃ is not a solution.

I still don't get how to use the Schrodinger equation to check if u2 and u3 are soltuions. But anyway, I think u2 is right. But u₃ = Ee^ik_ox. F is zero since there are no partciles hitting the barrier from the right.

Next, A + B = C + D and A - B = (ki/k0) (C - D).

Therefore, I can solve for C and D in terms of A and B.

Next, Ee^ik₀w = C - D
and (k0/ki) Ee^ik₀w = C + D

Therefore, Ee^ik₀w = (k0/ki) (A - B)
and Ee^ik₀w = (ki/k0) (A + B)

Dividing one equation by the other leads to B = A $$\frac{k^{2}_{0}-k^{2}_{i}}{k^{2}_{0}+k^{2}_{i}}$$.

Now, we need the ratio of the transmitted wavefunction amplitude to the incoming wavefunction amplitude i.e. E/A.

Am I right so far?

 

[fzero]

u₂ and u₃ must be solutions to the Schrodinger equation in the appropriate regions. You need to check this and, if necessary, modify them so that they are. Hint: u₃ is not a solution.

I still don't get how to use the Schrodinger equation to check if u2 and u3 are soltuions.

You practically did this in the first post, except that the calculation skipped computing

$$\frac{d^2}{dx^2} e^{\pm i k x}.$$

But anyway, I think u2 is right. But u₃ = Ee^ik_ox. F is zero since there are no partciles hitting the barrier from the right.

Next, A + B = C + D and A - B = (ki/k0) (C - D).

Therefore, I can solve for C and D in terms of A and B.

Next, Ee^ik₀w = C - D
and (k0/ki) Ee^ik₀w = C + D

Therefore, Ee^ik₀w = (k0/ki) (A - B)
and Ee^ik₀w = (ki/k0) (A + B)

Dividing one equation by the other leads to B = A $$\frac{k^{2}_{0}-k^{2}_{i}}{k^{2}_{0}+k^{2}_{i}}$$.

Now, we need the ratio of the transmitted wavefunction amplitude to the incoming wavefunction amplitude i.e. E/A.

Am I right so far?

Are you sure that both C and D are nonzero? Otherwise the boundary conditions look correct. The only other thing is that the transmission coefficient is usually expressed in terms of the probability densities, so

$$T = \frac{ |\psi_{\tex{transmit.}}|^2}{|\psi_{\tex{incident}}|^2},$$

so you're looking for the modulus squared of the quantity that you've computed.

 

[hasan_researc]

Thank you very much!

So, I understand that everything so far has been correct.

I'm pretty sure C and D are both non-zero, aren't they?

So, to move on, I have calculated E/A = $$\frac{2k^{2}_{0}}{k^{2}_{0}+k^{2}_{i}} e^{-ik_{0}w}$$.

This means that the transmission coefficient = square of the modulus of E/A = $$\frac{2k^{2}_{0}}{k^{2}_{0}+k^{2}_{i}}^{2}$$.

I hope this is all right.

I'd like to move on to the last part of the question:

" Find the ratio of the probability density at the point x = w/2 to the incoming wavefunction
probability density and state whether the magnitude of the ratio is greater
than or smaller than one.

Comment on the physical interpretation of the answer to part 6 above. "

This is my attempt at it.

$$\frac{ |\psi_{\tex{x = w/2}}|^2}{|\psi_{\tex{x = 0}} |^2} = \frac{ | Ce^{ik_{i}w/2} + De^{-ik_{i}w/2} |^2}{| A + B |^2}$$

I'm not sure what I should do next ?

 

[fzero]

Since the incoming wavefunction is only the term proportional to A, the "ratio of the probability density at the point x = w/2 to the incoming wavefunction
probability density" should be |E|^2/|A|^2. From the boundary conditions, you should also find that

$$|E|^2 = | Ce^{ik_{i}w/2} + De^{-ik_{i}w/2} |^2$$

once you simplify the expression on the right-hand side and use the relationships between the coefficients that you found earlier.

 

[hasan_researc]

Erm... I don't really understand why the numerator has to be the square of the modulus of E. I mean, we are evaluating the wavefunction at x = w/2, which means that we should use $$u = Ce^{ik_{i}x} + De^{-ik_{i}x}$$ for the numerator, shouldn't we?

 

[hasan_researc]

Or do you mean that we start with $$u = Ce^{ik_{i}x} + De^{-ik_{i}x}$$ in the numerator and find that it equal to E at x = w/2?

 

[hasan_researc]

Ok... I have thought about it and I think I'm pretty sure that's what you meant. Ok, so we find that the ratio is |E|^2/|A|^2. This means that the ratio is $$\frac{2k^{2}_{0}}{k^{2}_{0}+k^{2}_{i}}^{2}$$. This is less than 1. (Am I right?)

To finish off, I have asked to comment on the physical interpretation of the previous answer. Well, I'm thinking that the since the ratio half-way into the barrier is the same as the transmission coefficient, there is no chance of a particle returning to x < 0 if it crosses x =w/2 and moves to the right. Am I right?

 

[fzero]

No it was a mistake on my part, I didn't read carefully enough and thought they were talking about x = w.

My point about A vs A+B stands, so I think you want to simplify

$$\frac{ | Ce^{ik_{i}w/2} + De^{-ik_{i}w/2} |^2}{| A |^2} = \frac{ ( Ce^{ik_{i}w/2} + De^{-ik_{i}w/2} )( Ce^{-ik_{i}w/2} + De^{ik_{i}w/2} )}{| A |^2} .$$
You'll probably want to express the result in terms of trig functions.

 

[hasan_researc]

Well, we know from before that $$k_{i}w = \pi$$. Therefore

$$\frac{ |\psi_{\tex{x = w/2}}|^2}{|\psi_{\tex{x = 0}} |^2} = \frac{ | Ce^{ik_{i}w/2} + De^{-ik_{i}w/2} |^2}{| A + B |^2} = \frac{ | Ce^{i\pi/2} + De^{-i\pi/2} |^2}{| A |^2} = \frac{ | i (C - D) |^2}{| A |^2} = \frac{ | Ee^{ik_{0}w}e^{i\pi/2} |^2}{| A |^2} = \frac{ | E |^2}{| A |^2}$$.

Am I right?

 

[fzero]

Well, we know from before that $$k_{i}w = \pi$$. Therefore

$$\frac{ |\psi_{\tex{x = w/2}}|^2}{|\psi_{\tex{x = 0}} |^2} = \frac{ | Ce^{ik_{i}w/2} + De^{-ik_{i}w/2} |^2}{| A + B |^2} = \frac{ | Ce^{i\pi/2} + De^{-i\pi/2} |^2}{| A |^2} = \frac{ | i (C - D) |^2}{| A |^2} = \frac{ | Ee^{ik_{0}w}e^{i\pi/2} |^2}{| A |^2} = \frac{ | E |^2}{| A |^2}$$.

Am I right?

That looks right, but I was trying to check the physical interpretation and found that your

$$\frac{E}{A} = \frac{2k^{2}_{0}}{k^{2}_{0}+k^{2}_{i}} e^{-ik_{0}w}$$

has a mistake. I find

$$\frac{E}{A} = \frac{2k_{0} k_i}{k^{2}_{0}+k^{2}_{i}} e^{-ik_{0}w}.$$

This is pretty important if you want to compare the probability of transmission to the probability of reflection.

 

[hasan_researc]

Thanks!

So how do I actually physically interpret the result?

 

[hasan_researc]

And as an aside, I got your result in the end, but I don't understand why we have to compare the probability of transmission to the probability of reflection. I mean, A does not signify reflection. B does.

 

[fzero]

And as an aside, I got your result in the end, but I don't understand why we have to compare the probability of transmission to the probability of reflection. I mean, A does not signify reflection. B does.

The particle is either reflected or transmitted and there's an associated mathematical identity between the probabilities of transmission and reflection that has to be satisfied. I was just using that as a consistency check. I'm surprised that it isn't part of the problem.