144 3 Quantum Mechanics – II
conditions to sketch the form ofψ(x) in the region aroundx =0for
the cases (i) and (ii).3.42 A steady stream of particles with energyE(>V 0 ) is incident on a potential
step of heightV 0 as shown in Fig. 3.3.
The wave functions in the two regions are given by
ψ 1 (x)=A 0 exp(ik 1 x)+Aexp(−ik 1 x)
ψ 2 (x)=Bexp(ik 2 x)
Write down expressions for the quantitiesk 1 andk 2 in terms ofEandV 0.
Show that
A=[
k 1 −k 2
k 1 +k 2]
A 0 andB=[
2 k 1
k 1 +k 2]
A 0
and determine the reflection and transmission coefficients in terms ofk 1 and
k 2.
IfE= 4 V 0 /3 show that the reflection and transmission coefficients are 1/9
and 8/9 respectively.
Comment on whyA^2 +B^2 is not equal to 1.Fig. 3.3Potential step
3.43 (a) What boundary conditions do wave-functions obey?
A particle confined to a one-dimensional potential well has a wave-function
given by
ψ(x)=0forx<−L/2;
ψ(x)=Acos
(
3 πx
L)
for−L
2
≤x≤L
2
;
ψ(x)=0forx>L
2
(b) Sketch the wave-functionψ(x).
(c) Calculate the normalization constantA.
(d) Calculate the probability of finding the particle in the interval−L 4 <x<
L
4.
(e) By calculating d^2 ψ/dx^2 and writing the Schrodinger equation as
(
−^2
2 m)(
d^2 ψ
dx^2)
=Eψ.show that the energyEcorresponding to this wave-function is^9 π(^2) 2
2 mL^2.
3.44 (a) Sketch the one-dimensional “top hat” potential (1)V=0forx<0; (2)
V=W=constant for 0≤x≤L;(3)V=0forx>L.
(b) Consider particles, of massmand energyE<Wincident on this potential
barrier from the left (x<0). Including possible reflections from the barrier