1000 Solved Problems in Modern Physics

(Tina Meador) #1

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
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