Rectangular Potential and Constraints

[Kreizhn]

Hey all,

A friend asked me for help the other day on his QM homework. The problem regards a rectangular potential
$$U(x) = \begin{cases} V_0 & -a \leq x \leq a \\ 0 & \text{otherwise} \end{cases}, \qquad E<V_0$$
I thought about this for a while and checked a few textbooks. If we solve this in a piecewise form, we get
$$\psi(x) = \begin{cases} L_1 e^{ik_Lx} + L_2 e^{-ik_Lx} & x < -a \\ C_1 e^{kx} + C_2 e^{-kx} & -a < x < a \\ R_1 e^{ik_Rx} + R_2 e^{-ik_Rx} & x > a \end{cases}$$
Now by demanding that $\psi(x)$ be continuous and have continuous derivatives, we get 2 conditions from $x=-a$ and 2 from $x=a$ for a total of 4 conditions. Normalization gives us a 5th condition, but we need 6 in total. Now according to the textbooks, we can just set $R_2 = 0$. My question is, what is the motivation that allows us to set $R_2 = 0$?

Edit: Sorry, I perhaps should have been more explicit. I hope it's clear I'm talking about solving the time dependent Schrödinger equation, and $k_L, k_R$ are appropriately defined constants. I didn't think they were important but meant to include them originally.

 

[LostConjugate]

Edited out my first paragraph, I noticed it was incorrect.

I suppose if in the book they are assuming the incoming particles are only coming from the negative x side, then you could set $$R_2$$ to 0. It is the probability amplitude a particle will reflect off the potential coming from the positive side. If there is no particles heading in that direction, there is no probability.

 

[dextercioby]

Actually a full analysis discusses the solutions including the time dependency. That analysis will include the notions of incoming probability wave and outgoing probability wave. When these 2 are properly defined, 2 of the complex exponentials terms will vanish.

 

[LostConjugate]

which 2?

 

[dextercioby]

Both for the incoming wave and for the outgoing one the complex exponentials in t and x must have opposite signs (I assumed incoming wave from the left (-infty -> x) going to the right (x->+\infty)).

 

[Avodyne]

The condition $R_2=0$ is motivated by saying that the $L_1$ term corresponds to a wave incident from the left, the $L_2$ term corresponds to a wave that was reflected from the potential, and the $R_1$ term corresponds to a wave that was transmitted through the potential. We would want the $R_2$ term only if we had an incident wave from the right as well as the left.

With $R_2=0$, the reflection probability is $|L_2/L_1|^2$, and the transmission probability is $|R_1/L_1|^2$. These should sum to 1, and will if you solve the problem correctly.

 

[LostConjugate]

Doesn't the transmission probability have to account for the difference in wave number k (difference in momentum)?

 

[Kreizhn]

Thanks for the replies everyone. Sorry I'm so late in replying myself, my email screwed up :S