100 2 Quantum Mechanics – I
2.65 TheJ= 0 →J=1 rotational absorption line occurs at wavelength 0.0026
in C^12 O^16 and at 0.00272 m in CxO^16. Find the mass number of the unknown
Carbon isotope.
2.66 Assuming that the H^2 molecule behaves like a harmonic oscillator with force
constant of 573 N/m. Calculate the vibrational quantum number for which the
molecule would dissociate at 4.5 eV.
2.2.7 Commutators .....................................
2.67 (a) Show thateipα/xe−ipα/=x+α
(b) IfAandBare Hermitian, find the condition that the productABwill be
Hermitian
2.68 (a) IfAis Hermitian, show thateiAis unitary
(b) What operator may be used to distinguish between
(a)eikxande−ikx(b) sinaxand cosax?
2.69 (a) Show that exp (iσxθ)=cosθ+iσxsinθ
(b) Show that(d
dx)†
=−ddx2.70 Show that
(a) [x,px]=[y,py]=[z,pz]=i
(b) [x^2 ,px]= 2 ix
2.71 Show that a hermitian operator is always linear.
2.72 Show that the momentum operator is hermitian
2.73 The operators( PandQcommute and they are represented by the matrices
12
21
)
and(
32
23
)
. Find the eigen vectors ofPandQ. What do you notice
about these eigen vectors, which verify a necessary condition for commuting
operators?
2.74 An operatorAˆis defined asAˆ=αxˆ+iβpˆ, whereα,βare real numbers
(a) Find the Hermitian adjoint operatorAˆ†
(b) Calculate the commutators [Aˆ,xˆ], [Aˆ, Aˆ] and [Aˆ,Pˆ]
2.75 A real operatorAsatisfies the lowest order equation.
A^2 − 4 A+ 3 = 0
(a) Find the eigen values ofA(b) Find the eigen states ofA(c) Show thatA
is an observable.2.76 Show that (a) [x,H]=iμp(b) [[x,H],x]=
2
μwhereHis the Hamiltonian.
2.77 Show that for any two operatorsAandB,
[A^2 ,B]=A[A,B]+[A,B]A