leothelion
Oct6-08, 09:49 PM
Solved... i think. It's just Psi(x) = B sin kx
1. The problem statement, all variables and given/known data
Consider a free particle psi(x) = A*e^(ikx) approaching an infinite barrier from the left:
V = 0, x < 0 and V = oo, x >= 0. For this problem use only the time-independent
Schrodinger Equation.
a. Find the probability of being reflected from the barrier (recall that energy
must be conserved).
b. Find the total wavefunction Psi(x) for x < 0.
2. Relevant equations
Schroedinger's equation.
3. The attempt at a solution
Okay, solving the wave equation gives the general solution
psi(x, 0) = A e^(ikx) + B e^(-ikx).
We know that Psi(x, 0) = 0 when x >= 0, since E < V = oo for all values of E. Part a asks to find the reflection coefficient using conservation of energy, but I did it this way:
Psi must be continuous at x = 0. Thus A + B = 0, implying that A = -B. Thus the reflection coefficient is 1. This also reduces psi(x, 0) = A sin kx.
However, I'm a little confused as to how to find Psi(x). My original work was that Psi(x) = integral(-oo to oo) phi(k) A sin kx dk
but my professor said that due to the conservation of energy, there are only so many allowed energy states. I'm a little confused about this. Is Psi(x, 0) = A e^(ikx) - A e^(-ikx)?
Any hints would be greatly appreciated. Thanks!
1. The problem statement, all variables and given/known data
Consider a free particle psi(x) = A*e^(ikx) approaching an infinite barrier from the left:
V = 0, x < 0 and V = oo, x >= 0. For this problem use only the time-independent
Schrodinger Equation.
a. Find the probability of being reflected from the barrier (recall that energy
must be conserved).
b. Find the total wavefunction Psi(x) for x < 0.
2. Relevant equations
Schroedinger's equation.
3. The attempt at a solution
Okay, solving the wave equation gives the general solution
psi(x, 0) = A e^(ikx) + B e^(-ikx).
We know that Psi(x, 0) = 0 when x >= 0, since E < V = oo for all values of E. Part a asks to find the reflection coefficient using conservation of energy, but I did it this way:
Psi must be continuous at x = 0. Thus A + B = 0, implying that A = -B. Thus the reflection coefficient is 1. This also reduces psi(x, 0) = A sin kx.
However, I'm a little confused as to how to find Psi(x). My original work was that Psi(x) = integral(-oo to oo) phi(k) A sin kx dk
but my professor said that due to the conservation of energy, there are only so many allowed energy states. I'm a little confused about this. Is Psi(x, 0) = A e^(ikx) - A e^(-ikx)?
Any hints would be greatly appreciated. Thanks!