Description
1 Perfect gas of fermions (spin 1/2)
Consider a free gas of N non-relativistic electrons, whose single particle wavefunctions are
described by plane-waves normalized by the volume of the space.
The kinetic energy of the
system given by:
K = −
~
2
2m
ˆ
d
dx Ψ
†
(x)∇2Ψ(x),
where
Ψ(x) =
ψ↑(x)
ψ↓(x)
is a two component spinor, with ψσ (σ =↑, ↓) representing the annihilation field operator for
up/down electrons, and d is the dimensionality of the space.
a) Write the Kinetic energy in terms of creation and annihilation operators in the momentum space, a
†
σk
and aσk.
b) Write down the momentum operator P in terms of Ψ†
(x) and Ψ(x) field operators and
then rewrite it in terms of the creation and annihilation operators in the momentum space
a
†
σk
and aσk.
c) The ground state of the system is described by the state:
|F Si =
Y
α≤N
c
†
α|0i,
where cα ≡ aσ,k, with the notation α ≤ N meaning |k| ≤ kF , with kF the radius of the Fermi
Surface (FS) and σ =↑, ↓ for momentum states inside the FS. This state has the property that
aσ,k|F Si = 0 for |k| > kF
a
†
σ,k
|F Si = 0 for |k| < kF
Assuming d = 2 (2D electron gas), compute the total energy of the system at the ground state,
hKiF S and the total momentum, hPiF S. Interpret your result.
d) Suppose now that a uniform magnetic field B is turned on. The total Hamiltonian
becomes K + HB, where
HB = −µBS · B,
where µB is the Zeman coupling of the electronic spin to the magnetic field and
S =
~
2
ˆ
d
dxΨ
†
(x)~σΨ(x)
is the spin operator written in terms of field operators, with ~σ = (σx, σy, σz) as Pauli matrices.
Write HB in terms of a
†
σk
and aσk operators and then compute the total energy of the system
(assume d = 2) for each spin in separate. Explain what happens with the ground state (Hint:
assume B along the z-axis).
2 Hydrogen atom
Assume that an electron occupies a p level, which is degenerate among the three states |n, j =
1, mi, with m = ±1, 0. The electron is subjected to a perturbation V (r) = α(x
2 − y
2
).
a) Write the perturbation V (r) in terms of spherical tensors of rank 2.
b) Using the Wigner-Eckart theorem, write the perturbation matrix in the |n, j = 1, mi
basis.
c) Find the splitting of the p energy levels in first order in perturbation theory. You don’t
have to solve the integrals.
d) Assume now that the degeneracy of the p levels is lifted by a strong magnetic field B
pointing along the z axis, with the Hamiltonian (ignore spin effects)
HZ = µBBzm.
If µBBz α, calculate the energy correction to the |n, j = 1, mi states (in the presence of
the strong field) due to the perturbation V (r) in lowest order in perturbation theory where
the result is non-zero.
3 1D Harmonic oscillator
A particle with mass m is subjected to an harmonic potential
V (x) = 1
2
mω2x
2
.
The potential is then perturbed by an anharmonic force with potential
δV (x) = λ sin κx.
a) Find the corrected ground state ket in leading order in perturbation theory.
b) Using your result in a), calculate the expectation value of the position operator in the
corrected ground state. Hint: Use the identity:
e
A+B = e
Ae
Be
−[A,B]/2
.
c) Now assume that κx 1, such that δV (x) ≈ λκx. Use second order perturbation
theory to calculate the corrected energy to the ground state.
