# Annoying algebra

• May 19th 2013, 05:57 PM
froodles01
Annoying algebra
It's late and I really am at an end on this, I just can't seem to get it.

Note: A+B = C+D

So, I need to find B in terms of A.

A+B / A-B = k1/k2 * C+D / C-D

cross multiply
(A+B)k2 / (A-B)k1 = C+D / C-D
but A+B = C+D
so
(A+B)k2 / (A-B)k1 = A+B / C-D

=> (A+B)k2 / (A-B)k1) / (A+B) = C-D

(A+B)k2 / (A-B)k1 * (A+B) = C-D

Aagh!
• May 19th 2013, 11:52 PM
chiro
Re: Annoying algebra
Hey froodles01.

Just for clarification can you tell us all the information you started off with? You said A+B=C+D which is one piece of information: is there any more?

Since you have four variables to start off with. it means you will need one more two get a result in two variables. Can we assume that your second result involving A-B is given?

(Sorry if it seems stupid to ask but you said assume one thing and then you state something completely different).
• May 20th 2013, 01:07 AM
froodles01
Re: Annoying algebra
Hi.
yes, there's always more . . . .
A steady beam of particles travels in the x-direction and is incident on a finite square barrier of height V0, extending from x =0 to x = L. Each particle in the beam has mass m and total energy E0 =2V0. Outside the region of the barrier, the potential energy is equal to 0.
In the stationary-state approach, the beam of particles is represented by an energy eigenfunction of the form

ψ(x)=

Ae^ik1x + Be^−ik1x for x< 0
Ce^ik2x + De^−ik2x for 0 ≤ x ≤ L
Fe^ik1x for x>L

where A, B, C, D and F are constants and k1 and k2 are wave numbers appropriate for the different regions.
Express the wave numbers k1 and k2 in terms of V0, m and n. (which I have done - hooray [k1 = √2mE0/ħ and k2 = √2m(E0-V0)/ħ])

Now consider the special case where k2L = pi/2 (which does not correspond to a transmission resonance). Use your answer to part (b) to show that
A+B / A-B = k1/k2 * C+D / C-D

Use your answer for part (a) and the above equation to find B in terms of A. Hence calculate the reflection coefficient, R, of the beam and deduce the value of the transmission coefficient, T.

Useful equations include
A + B = C + D and
k1A – k1B = k2C – k2D
Ce^ik2L + De^-ik2L = Fe^ik1L
and k2Ce^ik2L – k2De^-ik2L = k1Fe^ik1L
k2L = pi/2

. . . . .well, you did ask!

I should be able to do this, but left myself too little time & now I'm too thick to think enough.