Potential Barrier (quantum mechanics)

[roam]

Homework Statement

I need help with part (iii) and (ii) of the follwoing problem:

http://img228.imageshack.us/img228/516/79620850.jpg

Homework Equations

From my notes transmission coefficient for E<U is:

$$T(E)= \left[ 1+\frac{1}{4} \left( \frac{U^2}{E(U-E)} \right) sinh^2 (\alpha L) \right]^{-1}$$

And for E>U it is:

$$T(E)= \left[ 1+\frac{1}{4} \left( \frac{U^2}{E(E-U)} \right) sin^2 (k' L) \right]^{-1}$$

Where $$k'=\frac{\sqrt{2m(E-U)}}{\hbar}$$ and $$\alpha =\frac{\sqrt{2m(U-E)}}{\hbar}$$.

The Attempt at a Solution

So for part (ii). I used the first equation and got a value of 4.59x10^-4. This doesn't seem like a correct number for the number of neutrons. So do I have to multiply this with the total number of neutrons?

For (iii) I am confused and I don't know which equation to use. We know the equation when E<U and when E>U, but what exactly is the equation for when E=U?

I don't quite understand the hint, does this mean we need the following equation:

$$T(E)= \left[ 1+\frac{1}{4} \left( \frac{U^2}{E(U-E)} \right) \alpha^2 L^2 \right]^{-1}$$

Is that right?

 

[vela]

Homework Statement

I need help with part (iii) and (ii) of the follwoing problem:

http://img228.imageshack.us/img228/516/79620850.jpg

Homework Equations

From my notes transmission coefficient for E<U is:

$$T(E)= \left[ 1+\frac{1}{4} \left( \frac{U^2}{E(U-E)} \right) sinh^2 (\alpha L) \right]^{-1}$$

And for E>U it is:

$$T(E)= \left[ 1+\frac{1}{4} \left( \frac{U^2}{E(E-U)} \right) sin^2 (k' L) \right]^{-1}$$

Where $$k'=\frac{\sqrt{2m(E-U)}}{\hbar}$$ and $$\alpha =\frac{\sqrt{2m(U-E)}}{\hbar}$$.

The Attempt at a Solution

So for part (ii). I used the first equation and got a value of 4.59x10^-4. This doesn't seem like a correct number for the number of neutrons. So do I have to multiply this with the total number of neutrons?

What does T(E) represent? If you understand that, it should be clear how to proceed.

For (iii) I am confused and I don't know which equation to use. We know the equation when E<U and when E>U, but what exactly is the equation for when E=U?

I don't quite understand the hint, does this mean we need the following equation:

$$T(E)= \left[ 1+\frac{1}{4} \left( \frac{U^2}{E(U-E)} \right) \alpha^2 L^2 \right]^{-1}$$

Is that right?

Yes. Substitute in for $\alpha$ now.

 

[roam]

What does T(E) represent? If you understand that, it should be clear how to proceed.

Thank you for your reply. I believe T(E) represents the probability of the particles being transmitted relative to the total incident particles. So isn't it correct then to multiply it by the total number of particles to see how many are transmitted?

In that case I will have (10⁶)x(4.59x10^-4)=459

Yes. Substitute in for $\alpha$ now.

But doesn't $\alpha$ equal to 0 when U=E?

$\alpha = \frac{\sqrt{2(1.675\times 10^{-27})(4MeV-4MeV)}}{1.054 \times 10^{-34}} = 0$

This will make the whole T(E) expression equal to 1, so that means that all the particles get transmitted? Is that right or did I make a mistake?

 

[vela]

Thank you for your reply. I believe T(E) represents the probability of the particles being transmitted relative to the total incident particles. So is isn't it correct then to multiply it by the total number of particles to see how many are transmitted?

In that case I will have (10⁶)x(4.59x10^-4)=459

Exactly, T(E) is the fraction that gets transmitted.

But doesn't $\alpha$ equal to 0 when U=E?

$\alpha = \frac{\sqrt{2(1.675\times 10^{-27})(4MeV-4MeV)}}{1.054 \times 10^{-34}} = 0$

This will make the whole expression equal to 1, so that means that all the particles get transmitted? Is that right or did I make a mistake?

No, remember the expression for T(E) is valid for E<U, so U-E isn't 0. Do the substitution and simplify, and then take the limit as E approaches U.

 

[roam]

Exactly, T(E) is the fraction that gets transmitted.

Thanks, so I guess it is correct to multiply T(E) by the total number of particles.

No, remember the expression for T(E) is valid for E<U, so U-E isn't 0. Do the substitution and simplify, and then take the limit as E approaches U.

I see. So here is what I did:

$\lim_{E \to U} \left( 1+ \frac{1}{4}\frac{U^2}{E(U-E)} . \frac{2m(U-E)}{\hbar^2} L^2 \right)^{-1}$

$\lim_{E \to U} \left( 1+ \frac{mU^2}{2 \hbar E} L^2 \right)^{-1}$

And since for small x we have (1+x)^-1=1-x,

$\lim_{E \to U} \left( 1- \frac{mU^2}{2 \hbar E} L^2 \right)^{-1} = \frac{mU^2}{2 \hbar U} L^2$

Is this correct now?

 

[vela]

No, you made algebra mistakes, and you shouldn't approximate using a series.

 

[roam]

Oops that was a typo, here is what I've got

$$\lim_{E \to U} \left( 1+ \frac{mU^2}{2 \hbar^2 E} L^2 \right)^{-1}$$

I have substituted the limits first and it is possible to take the inverse later:

$T(x)=\left( 1+ \frac{mU}{2 \hbar^2} L^2 \right)^{-1}$

Is that fine now?

 

[vela]

Looks fine.

 

[roam]

Alright, thank you so much for the help.