1000 Solved Problems in Modern Physics

2.3 Solutions 101

2.78 Show that (σ.A)(σ.B)=A.B+iσ.(A×B) whereAandBare vectors andσ’s
are Pauli matrices.

2.79 The Pauli matrixσy=

(

0 −i

i 0

)

(a) Show that the matrix is real whose eigen values are real.

(b) Find the eigen values ofσyand construct the eigen vectors.

(c) Form the projector operatorsP 1 andP 2 and show that

P 1 †P 2 =

(

00

00

)

,P 1 P 1 †+P 2 P 2 †=I

2.80 The Pauli spin matrices areσx =

(

01

10

)

,σy=

(

0 −i

i 0

)

,σz=

(

10

0 − 1

)

Show that (i)σx^2 =1 and (ii) the commutator [σx,σy]= 2 iσz.

2.81 The condition that must be satisfied by two operatorsAˆandBˆif they are to
share the same eigen states is that they should commute. Prove the statement.

2.2.8 Uncertainty Principle...............................

2.82 Use the uncertainty principle to obtain the ground state energy of a linear
oscillator

2.83 Is it possible to measure the energy and the momentum of a particle simulta-
neously with arbitrary precision?

2.84 Obtain Heisenberg’s restricted uncertainty relation for the position and momen-
tum.

2.85 Use the uncertainty principle to make an order of magnitude estimate for the
kinetic energy (in eV) of an electron in a hydrogen atom.
[University of London 2003]

2.86 Write down the two Heisenberg uncertainty relations, one involving energy
and one involving momentum. Explain the meaning of each term. Estimate
the kinetic energy (in MeV) of a neutron confined to a nucleus of diameter
10 fm.
[University of London 2006]

2.3 Solutions..................................................

2.3.1 deBroglieWaves ..................................

2.1 (a)E=hf=hλ/c
(b) Each photon carries an energy
Eγ=

mπc^2

2

=

135

2

= 67 .5MeV