Physics IV
ISI B.Math
Final Exam : April 26, 2023
Total Marks: 50
Time : 3 hours
Answer all questions
1. (Marks : 7 + 3 = 10 )
(a) A mass mmoving at speed 4
5ccollides with another mass mat rest. The collision produces a
photon with energy Etraveling perpendicular to the original direction, and a mass Mtraveling in
another direction as shown in the figure. In terms of Eand m,what is M? What is the largest
value of E( in terms of m) for which this setup is possible ?
(b) Write down a relativistic expression for the kinetic energy of a particle of mass mand speed v.
Show that it reduces to the usual Newtonian expression in the appropriate limit.
2. (Marks : 3 + 3 + 4 + 2 = 12)
The annihilation operator for the harmonic oscillator in 1-d is defined as ˆa=qmω0
2¯hˆx+iˆp
mω0
where mis the mass and ω0is the angular frequency of the harmonic oscillator
(a) If ˆ
His the Hamiltonian for the above one dimensional harmonic oscillator and ˆ
N= ˆa†ˆais the
number operator, show that ˆ
Nand ˆ
Hhave simultaneous eigenstates. Find the eigenvalues of the
energy in terms of the eigenvalues nof the number operator where n= 0,1,2···
(b) Given that |n > is a normalized eigenstate corresponding to the eigenvalue nof the Hamiltonian
of the harmonic oscillator where nis an integer and ˆa|n >=√n|n−1>and ˆa†|n >=√n+ 1|n+1 >,
Find <ˆ
x3>for the nth energy eigenstate of the harmonic oscillator.
(c) Show that the average kinetic energy < T > is equal to the average potential energy < V > for
any energy eigenstate of the harmonic oscillator
(d) In the case of the 1-d harmonic oscillator the energy levels are non-degenerate, i.e, an energy
eigenvalue is associated with a unique eigenstate. Now if we generalize to three dimensions and
consider the potentials i) V(x, y, z) = 1
2mω2(x2+y2+z2) and (ii) 1
2m(ω12x2+ω22y2+ω32z2) of
the three dimensional isotropic and anisotropic harmonic oscillators, discuss whether the feature of
nondegeneracy persists. You do not have to solve the entire problem in three dimensions all over
again, but just have to give a plausible argument.
3. (Marks : 3 + 3 + 4 = 10 )
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