3.2 Problems 143
Fig. 3.2Bound states in a
square well potential
3.36 In Problem 3.25 express the normalization constantAin terms ofα,βanda.
3.37 A particle of massmis trapped in a square well of widthLand infinitely deep.
Its normalized wave function within the well for thenth state is
ψn=(
2
L
) 1 / 2
sin(nπx
L)
(a) Show that its mean position isL/2 and the variance is
(
L^2
12)(
1 −
6
n^2 π^2)
(b) Show that these expectations are in agreement with the classical values
whenn→∞.3.38 The quantum mechanical Hamiltonian of a system has the form
H=(−^2 / 2 m)∇^2 +ar^2(
1 −
5
6
sin^2 θcos^2 φ)
Find the energy eigen value of the two lowest lying stationary states.3.39 (a) Write down the three-dimensional time-independent Scrodinger equation
in Cartesian co-ordinates. By separating the variables,ψ (x,y,z) =
X(x)Y(y)Z(z), solve this equation for a particle of massmconfined to
a rectangular box of sidesa, bandc, with zero potential inside.
(b) Show that the particle has energy given by
E=(
^2
8 m)[
n^2 x
a^2+
n^2 y
b^2+
n^2 z
c^2]
3.40 In Problem 3.39, consider the special case of a cubea=b=c.Drawupa
table listing the first six energy levels, stating the degeneracy for each level.
3.41 A particle of massmis moving in a region where there is a potential step at
x=0:V(x)=0forx<0 andV(x)=U 0 (a positive constant) forx≥ 0
(a) Determineψ(x) separately for the regionsx0 andx0 for the cases:
(i) U 0 <E
(ii) U 0 >E.(b) Write down and justify briefly the boundary conditions thatψ(x) must sat-
isfy at the boundary between the two adjacent regions. Use these