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Model Question: Third Semester M.Sc. (Physics)

1) The document is a model exam paper for the subject Quantum Field Theory. It contains 7 questions worth a total of 45 marks. 2) The first two questions relate to the Dirac formulation of the hydrogen atom and solving the Klein-Gordon equation for a particle incident on an electrostatic potential barrier. 3) The remaining questions cover topics such as interacting boson systems, Feynman propagators, gauge transformations in QED, and charge conjugation properties of relativistic particles.

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Sagar Rawal
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864 views1 page

Model Question: Third Semester M.Sc. (Physics)

1) The document is a model exam paper for the subject Quantum Field Theory. It contains 7 questions worth a total of 45 marks. 2) The first two questions relate to the Dirac formulation of the hydrogen atom and solving the Klein-Gordon equation for a particle incident on an electrostatic potential barrier. 3) The remaining questions cover topics such as interacting boson systems, Feynman propagators, gauge transformations in QED, and charge conjugation properties of relativistic particles.

Uploaded by

Sagar Rawal
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Model Question: Third Semester M.Sc.

(Physics)
Subject: Quantum Field Theory (PHY651) Duration: 2 hrs
Full Marks: 45 Pass Marks: 22.5

Attempt all questions.

1. Apply Dirac formulation to the Hydrogen atom and find expression for energy eigenvalue. Discuss the
conditions under which Sommerfeld’s fine structure formula can be obtained. [10]

2. Set up Klein Gordon equation and find probability current density and probability density. Calculate
transmission coefficient for a Klein Gordon particle with mass m and charge q having energy E that is
incident on electrostatic potential:
0; 𝑥≶0
𝑉(𝑥) = {
𝑉; 0 ≤ 𝑥 ≤ 𝑎
Discuss the nature of the solution when (a) 𝑒𝑉 > 𝐸 − 𝑚, (b) 𝑒𝑉 > 𝐸 + 𝑚, and (c) 𝐸 − 𝑚 < 𝑒𝑉 < 𝐸 +
𝑚. Discuss the condition when K-G particle is transmitted by tunneling. [10]
OR
𝜇
Show that free Dirac Lagrangian ℒ0 = 𝜓(𝑖𝛾 𝜕𝜇 − 𝑚)𝜓 is invariant under C, P, and T operations,
separately.

3. Discuss the system of interacting Bosons. Show that a system of interacting helium atoms at T = 0 K
behaves as a collection of elementary excitations corresponding to non-interacting quasi-particles
(quanta) of momentum ℏk and energies E(k), given by
2 1/2
2𝑣𝑛ℏ2 𝑘 2 ℏ2 𝑘 2
𝐸(𝑘) = [ 𝑚
+ ( 2𝑚 ) ] .
Where the symbols have their usual meaning. [5]

4. Find the expression for Feynman propagator for Klein-Gordon particle as,
𝑑4𝑝 𝑖
𝐷𝐹 (𝑥 − 𝑦) = ∫ 𝑒 𝑖𝑝.(𝑥−𝑦)
(2𝜋)4 𝑝2 − 𝑚2 + 𝑖𝜀
Where the symbols have their usual meaning. [5]
OR
Discuss determinant, trace and order of x, y, z and  matrices introduced by Dirac in his
Hamiltonian. How these properties differ from the properties of Pauli spin matrices? Set up x, y, z
and  matrices in terms of Pauli spin matrices.

5. What do you mean by Gauge Covariant Derivative? Show that QED lagrangian is invariant under Gauge
Transformation. [5]

6. Show that the Dirac field bilinear


(a) 𝜓𝜓 transforms like a scalar.
(b) 𝜓𝛾 𝜇 𝜓 transforms like a vector. [5]

7. Discuss charge conjugation property of charged, zero spin, free relativistic particle (or KG particle).
[5]

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