# Seemingly difficult complex arithmetic problem

1. Jan 30, 2010

### PhysicsMark

1. The problem statement, all variables and given/known data
Transmission of a quantum mechanical wave past a one-dimensional square well results in the following expressions relating initial to final wave amplitudes:

$$A= (cos(2ka)-{i\epsilon\over2}sin(2ka))e^{2ia\lambda}F+{i\eta\over2}sin(2ka)G$$
$$B= {-i\eta\over2}sin(2ka)F+(cos(2ka)+{i\epsilon\over2}sin(2ka))e^{-2ia\lambda}G$$

$$\epsilon={\lambda\over(k)}+{k\over\lambda}$$
$$\eta={\lambda\over(k)}-{(k)\over\lambda}$$

For G=0 (this represents an initial beam coming in only from the left) the "transmission" and "reflection" coefficients are defined respectively as:
$$T=|{F\over(A)}|^2$$
$$R=|{B\over(A)}|^2=1-T$$

Show that

$$T={1\over(cos^2(2ka)+{\epsilon^2\over4}sin^2(2ka))}$$

and

$$R={({\epsilon^2\over4}-1)sin^2(2ka)\over1+({\epsilon^2\over4}-1)sin^2(2ka)}$$
2. Relevant equations
See above

3. The attempt at a solution

This is a "challenge" problem from my mathematical methods in physics class. I have faith in this class and I would like to ask for a hint. What I mean is that I believe this problem will give me greater insight into complex arithmetic as the other challenge problems have. I worked with my group members for a few hours last night and didn't get anywhere. Mainly we kept trying to re-write things in different terms to see if we could reach the desired result. I believe that we are not thinking about the "spirit" (as my professor calls it) of the question. That is we are not thinking about how this question relates to complex arithmetic. Any takers?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited: Jan 31, 2010
2. Jan 31, 2010

### andylu224

It only requires some very basic trigonometric identities and properties of moduli.

Several Hints:

To derive T, 1) e-i(theta) = cos(theta) - i sin(theta)
2) How does one divide 2 complex numbers (ie. (x1 + iy1)/(x2 + iy2) )?
3) multiplicative property of moduli: |zw| = |z| x |w|
where z & w are complex

R just requires the use of some simple trigonometric identities, involving the T term that was derived.

Good Luck!

3. Jan 31, 2010

### PhysicsMark

Since G=0 we can drop off a portion of Eq A and B, right?

Hint 1. I am aware of hint 1 but I do not see how it fits in.
Hint 2. To divide 2 complex numbers, it is best to rationalize the denominator by multiplying by the complex conjugate.
Hint 3. I also know about that property of moduli. I had to prove it in an earlier exercise.

I am not tying things together here...

Last edited: Jan 31, 2010
4. Jan 31, 2010

### PhysicsMark

Sorry to bump, but I wanted to bring this problem to a wider audience. Can anyone else offer a bit more insight to the problem?

5. Feb 1, 2010

### vela

Staff Emeritus
So you set G to 0 and solve for F/A, and you get

$$\left|\frac{F}{A}\right|=\frac{1}{|[\cos(2ka)-{i\epsilon\over2}\sin(2ka)]e^{2ia\lambda}|}$$

What are $|\cos(2ka)-i(\epsilon/2)\sin(2ka)|$ and $|e^{2ia\lambda}|$ equal to?