Calculating Energy Eigenvalues & Eigenfunctions for a 2D Particle

[Franco]

let's say..

there is a particle, with mass m, in a 2-dimensions x-y plane. in a region
0 < x < 3L ; 0 < y < 2L

how to calculate the energy eigenvalues and eigenfunctions of the particle?

thx

and.. 2nd question..

there is a particle of kinetic energy E is incident from the left on the potential barrier, height U, situated at the origin. The barrier is infinitely wide and E>U.

how to get an expression for the reflection coefficient R of the particle, as a function of the ratio ε= E/U
and how would the sketch look like?


i don't really know how to work them out...

 

[HallsofIvy]

Basically, you are talking about a "potential well" in which the energy is taken to be 0 inside the rectangle, infinite outside. You need to solve Schrödinger's equation with Energy 0 inside the rectangle. This turns out to be one of the few situations where it can be solved exactly. The solution consists of "standing waves" (just like waves on a rubber rectangular sheet fixed at the boundaries).

 

[Franco]

i don't understand when it comes into 2 dimension :(

 

[Franco]

can someone help me see if these look right?

π = pi

0 < x < 3L ; 0 < y < 2L
E1 + E2 = E
Eigenvalue :
ψ(x,y) = A sin [(n1 π x) / (3 L)] sin [(n2 π y) / (2 L)]
Eigenfunction :
E = E1 + E2 = [(n1^2 π^2 ħ^2) / (6 m L^2)] + [(n2^2 π^2 ħ^2) / (4 m L^2)]
E = [(π^2 ħ^2) / (2 m L^2)] * [(n1^2 / 3) + (n2^2 / 2)]

for the 3 lowest energy
E11 = [(π^2 ħ^2) / (2 m L^2)] * [(1^2 / 3) + (1^2 / 2)]
= (5 π^2 ħ^2) / (12 m L^2)
E12 = [(π^2 ħ^2) / (2 m L^2)] * [(1^2 / 3) + (2^2 / 2)]
= (7 π^2 ħ^2) / (6 m L^2)
E21 = [(π^2 ħ^2) / (2 m L^2)] * [(2^2 / 3) + (1^2 / 2)]
= (11 π^2 ħ^2) / (12 m L^2)

 

[danitaber]

and.. 2nd question..

there is a particle of kinetic energy E is incident from the left on the potential barrier, height U, situated at the origin. The barrier is infinitely wide and E>U.

how to get an expression for the reflection coefficient R of the particle, as a function of the ratio ε= E/U
and how would the sketch look like?


i don't really know how to work them out...

http://electron6.phys.utk.edu/qm1/modules/m2/step.htm

Nor do I, but here's a jumping point if it'll help. . . my brain doesn't work after 1:00am EST