Quantum Field Theory
Practice Question for Mid-Term
SN Long Questions
1 Set up covariant form of KG equation and find probability charge and current
density of KG particle in the presence of e.m. field.
2 Show that isospin states can be dealt under KG formulation. Set up KG
Hamiltonian and find probability and current density for KG particle.
3 Discuss charge conjugation property of charged, zero spin, free relativistic
particle (or KG particle) and show that KG formulation verify it.
4 Find the spectrum of the Klien-Gordon Hamiltonian assuming the field as an
independent harmonic oscillator. Finally show that Klien-Gordon particle
obey Bose Einstein Statistics.
5 Set up radial equation for a KG particle in a centrally symmetric potential and
solve it using coulomb field for hydrogen atom. Find the expression for
energy in terms of relativistic correction. Discuss its incompatibility with
experimental results.
6 Consider the field theory of a complex valued scalar field obeying Klein
Gordon equation. The action of the theory is
S=d4x( * )-m2*).
a) Find the conjugate momenta to (x) and *(x).
b) Show that the Hamiltonian is given by
H=d3x( *+*.+m2*).
c) Writing the result of (b) in terms of annihilation and creation operators,
show that the Hamiltonian can be written as
H=d3x Ep(apap + bpbp ).
d) Given that Q=d3x( *(t )-(t *)) show that
Q=d3x Ep(apap - bpbp ).
7 Discuss the field theory of complex valued scalar field obeying Klien-Gordon
equation. Show that Heisenberg equation of motion is indeed Klein-Gordon
equation in this field.
SN Short Questions
1 Set up Klien-Gordon equation and discuss the difficulties with its solutions.
2 Find the expression for current density and probability density for KlienGordon equation. Compare it with the current density and probability density
obtained from Schrodinger equation. What are inconsistencies? Discuss.
3 Show that casualty is maintained in Klien-Gordon theory.
4 Discuss Klien-Gordon field in the Heisenberg picture regading the duel
particles and wave interpretation of the quantum field.
3 The propagator of Klein Gordon field is given by,
iDF(x-y)=T{(x) (y)}= (x) (y) if t(y)<t(x)
iDF(x-y)=T{(x) (y)}= (y) (x) if t(x)<t(y)
Show that
4
5
6
7
iD+(x-y)=T{ (y) (x)}=<0/[(+)(y),(-)(x)]/0>
Starting from the above expression show that
iD(+)(x-y)=-i/2(2)3d3k(exp(-ik.(x-y))/(2k)
Using the results of ques 3, show that
iD(+)(x-y)= -i/(2)4d4k(exp(-ik.(x-y))/(k2-m2)
Also show that modifying (k2-m2) to (k2-m2+i) leads to Feynman propagator.
Explain.
†
Show that K-G Hamiltonian is not Hermitian. Find how 𝐻𝐾−𝐺 and 𝐻𝐾−𝐺
are
related.
Derive Yukawa potential from Klein-Gordon equation and discuss it. Describe
Yukawa potential for bound orbits and discuss the condition for Newtonian
limit.
Under what condition the K-G equation of this form
𝑚2 𝑐 2
[∎ + 2 ] |𝜓⟩ = 0
ℏ
represents free Schrodinger equation for spin less particles given by
𝑖ℏ
𝜕
𝜕𝑡
|𝜓⟩ = −
ℏ2
2𝑚
∇2 |𝜓⟩ . Explain it.
SN Numerical
1 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.
2 Show that the charge density of K-G particle within the barrier (region II)
0;
𝑥≶0
𝑉(𝑥) = {
𝑉; 0 ≤ 𝑥 ≤ 𝑎
′
2
is given by 𝜌 = 2(𝐸 − 𝑒𝑉)|𝜓𝐼𝐼 | < −2𝑚|𝜓𝐼𝐼 |2 . Discuss the possibilities of
creation and annihilation when 𝑒𝑉 > 𝐸 + 𝑚 inside the barrier.
3 Show that the current of K-G particle inside the barrier (region II)
0;
𝑥≶0
𝑉(𝑥) = {
𝑉; 0 ≤ 𝑥 ≤ 𝑎
′
2
is given by 𝐽 = 2𝑃|𝑇| .
4 Solve K-G equation for a square-well potential of the form
−𝑉 ; 𝑟 ≤ 𝑅
𝑉(𝑟) = { 0
0;
𝑟>𝑅
where V0 > 0 and R > 0.
5 Since a pion is bound by a scalar potential of the form
𝑈(𝑥) = 𝑉(𝑟) = −𝑉0 𝛿 3 (𝑟).
Solve K-G equation for the special case when the solution is static (i.e,
independent of time). Discuss the significance of your result.
6
Consider the Lagrangian
1
𝜓 [∎ + 𝑚2 ]𝜓
2
and compute the Euler-Lagrange equation of this Lagrangian and show that
it yields the K-G equation.
The plane wave solution for K-G particle in the box normalization is given as:
𝑢0
|𝜓⟩ = [ 𝑉 ] 𝑒𝑥𝑝[𝑖(𝑝. 𝑟 − 𝐸𝑡)]
0
By using normalization condition for negative energy states show that
|𝑢0− |2 = |𝑉0− |2 = −1.
𝐿=
7
12 Dec 2018