Question

Consider a particle moving in two spatial dimensions, subject to the following potential:
V (x, y) = (
0, 0 ≤ x ≤ L & 0 ≤ y ≤ H
∞, otherwise.
(a) Write down the time-independent Schr¨odinger equation for this case, and motivate
its form. (2)
(b) Let k
2 = 2mE/~
2 and rewrite this equation in a simpler form. (2)
(c) Use the method of separation of variables and assume that ψ(x, y) = X(x)Y (y).
Rewrite the equation in terms of X and Y . (2)
(d) Divide by XY and solve for X00/X.
(e) Define a separation constant λ and write down the general solution for X(x). (2)
(f) Apply the boundary conditions in the x-dimension and obtain Xn(x). (4)
(g) Write down the general solution for Y (y). (2)
(h) Apply the boundary conditions in the y-dimension and obtain Ym(x). (3)
(i) Normalise ψnm(x, y). Make use of the fact that x and y are independent and that
the two integrals may thus be solved independently. (2)
(j) From the definition of k and using λ, obtain the discrete energies Enm. If we
define Enm ≡ Ex + Ey, write down expressions for the latter two terms

Answer 1

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