QFT Numerical (SET A)
Klein Gordon Formulation
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)
is given by 𝐽′ = 2𝑃|𝑇|2 .
0;
𝑉(𝑥) = {
𝑉;
𝑥≶0
0≤𝑥≤𝑎
4. Solve K-G equation for a square-well potential of the form
where V0 > 0 and R > 0.
−𝑉 ; 𝑟 ≤ 𝑅
𝑉(𝑟) = { 0
0;
𝑟>𝑅
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. A pion of mass m is bounded by a scalar one-dimensional potential V(x) defined to be
Region I
Region II
Region III
𝑅<𝑥
0<𝑥<𝑅
𝑥<0
𝑉(𝑥) = 0
𝑉(𝑥) = −𝜇2 𝑉0
𝑉(𝑥) = ∞
(This could be a very rough model for pion inside a nucleus of radius R)
(a) Solve K-G equation in one space dimension for a positive energy ground state [Take
U(x) = V(x)] (b) Find the value of R such that the positive energy ground state has energy
𝐸 = 𝜇√1 − 𝑉0 /2.
Estimate the size of the pion cloud.
†
7. Show that K-G Hamiltonian is not Hermitian. Find how 𝐻𝐾−𝐺 and 𝐻𝐾−𝐺 are related.
8. Solve radial part of K-G equation for a particle under the influence of coulomb potential
𝜙=−
And find energy eigenvalue as
𝑍𝑒
𝑟
𝑟2
𝑟4
𝑛
3
𝐸 = 𝑚𝑐 [1 − 2 − 4 (
− ) + ⋯]
2𝑛
2𝑛 𝑙 + 1 4
2
Where the first term is rest mass energy, the second and third term represent
relativistic Rydberg energy and relativistic correction, respectively. Find the spread of
fine structure level and discuss with the experimental results in the spectrum of
Hydrogen.
2
9. 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
Explain it.
𝑖ℏ
𝜕
𝜕𝑡
|𝜓⟩ = −
ℏ2
2𝑚
∇2 |𝜓⟩ .
10. 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.
𝐿=
11. 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.
12. If you are given 𝜓 = 𝑢 + 𝑣, where u and v are two different states, then by using the
properties of isospin matrices show that
𝜓∇𝜓 ∗ = ⟨∇𝜓|𝜏3 (𝜏3 + 𝑖𝜏2 )|𝜓⟩
13. Use proper gauge condition to show that
14. Derive Yukawa potential from Klein-Gordon equation and discuss it. Describe Yukawa
potential for bound orbits and discuss the condition for Newtonian limit.