- #1

Kreizhn

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A friend asked me for help the other day on his QM homework. The problem regards a rectangular potential

[tex] U(x) = \begin{cases} V_0 & -a \leq x \leq a \\ 0 & \text{otherwise} \end{cases}, \qquad E<V_0 [/tex]

I thought about this for a while and checked a few textbooks. If we solve this in a piecewise form, we get

[tex] \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} [/tex]

Now by demanding that [itex] \psi(x) [/itex] be continuous and have continuous derivatives, we get 2 conditions from [itex] x=-a [/itex] and 2 from [itex] x=a [/itex] 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 [itex] R_2 = 0 [/itex]. My question is, what is the motivation that allows us to set [itex] R_2 = 0[/itex]?

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