Q.M3 Home work 13 Due date 20.2.15 1

Q.M3 Home work 13
Due date 20.2.15
1
The phenomenology of conventional superconductor can be described by the effective
Hamiltonian
X
X
2
(∆ck↓ ck↑ + ∆c†k↑ c†k↓ ) + ∆g
H=
ζk c†kσ ckσ −
kσ
k
where ∆ is real constant, and the electron field operator obey Fermionic anticommutation
relations ckσ c†k0 σ0 + c†k0 σ0 ckσ = δkk0 δσσ0
a) Show that the transformation
! † !
†
cos(θk ) sin(θk )
ck↑
αk↑
=
sin(θk ) − cos(θk )
αk↓
ck↓
preserves the anticommutation relation of the new field operators.
b) Applying this transformation, rewrite the Hamiltonian in term of the field operators
†
and αk↑
αk↑
c) Show that, by suitable choice of θk , the Hamiltonian takes the diagonal form.
2
Obtain eigenstates of the following Hamiltonian:
H = ~ωa† a + V a + V ∗ a†
for a complex parameter V using coherent states.
3
Consider a two dimensional lattice where each lattice point (i,j) is connected to four
nearest neighbours namely, (i + 1, j), (i − 1, j), (i, j + 1), (i, j − 1).
1
The Lagrangian on this lattice is define as,
X
X
t(c∗j ci + c∗i cj )
L=
c∗i i~ċi −
i
hi,ji
where the subscript hi, ji stands for summation over all nearest-neighbours on the lattice.
Here, t is parameter (not time)
a) Use the canonical commutation relation [ci , ∂∂L
c˙i ] = i~ and find the Hamiltonian
b) Show that the state
X †
ck eikx |0i
k
is en eigenstate of the Hamiltonian.
Assume one dimensional space.
4
Consider a coherent state of the photons in a particular momentum p = (0, 0, p) and
helicity +1 1 state
|f i = e−f
∗ f /2
†
ef a+ (~p) |0i
Use the mode expansion of the Maxwell field:
r
2π~c2 X 1 X i
i
( (~
p)a± (~
p)ei~p~x/~ + i± (~
p)∗ a†± (~
p)e−i~p~x/~ )
A (~x) =
√
L3
ωp ± ±
p
~
r
i
Ȧ (~x) =
X
2π~c2 X
√
(−i ωp )
(i± (~
p)a± (~
p)ei~p~x/~ − i± (~
p)∗ a†± (~
p)e−i~p~x/~ )
3
L
±
p
~
a) Show that the Hamiltonian of photons is:
Z
XX
1 ~2 ~2
H = d~x (E
+B )=
c|~
p|(a†λ (~
p)aλ (~
p) + 12 )
8π
p
~
(1)
λ
Ignore the zero point energy term below.
b) Show that the Schroedinger equation
∂
i~ ∂t
|f i = H|f i
has a solution |f, ti = |f e−ic|~p|t/~ i
~ x)|f, ti. You can see that
c) Calculate the expectation value of the Maxwell field hf, t|A(~
this state describes a classical electromagnetic wave such as laser
1
The photon has two possible helicity. These two possible helicities, called right-handed and lefthanded, correspond to the two possible circular polarization states of the photon. Where helicity is
the projection of the spin onto the direction of momentum.
2
5
A polar representation of the creation and annihilation operators for a simple harmonic
oscillator can be introduced as
√
a = N + 1eiφ
√
a† = e−iφ N + 1
The operators N and φ are assumed to be Hermitian.
a) Starting from the commutation relation [a, a† ] = 1, show that
[eiφ , N ] = eiφ
[e−iφ , N ] = −e−iφ
Similarly, show that
[cos(φ), N ] = i sin(φ)
[sin(φ), N ] = −i cos(φ)
b) Calculate the matrix elements
hn|e±iφ |n0 i
hn| cos(φ)|n0 i
hn| sin(φ)|n0 i
where |ni is the ”n” energy state
c) Write down the Heisenberg uncertainty relation between the operators N and cos(φ)
(namely (∆N )2 (∆ cos(φ))2 ). Also compute again the Heisenberg uncertainty relation
between the operators N and cos(φ) involved for the state
|ψi = (1 − |c|)1/2
∞
X
cn |ni
n=0
where c is a complex parameter. Show that the resulting inequality is always true.
d) Consider the coherent state
|zi = e−|z|
2 /2
∞
X
zn
√
|ni
n!
n=0
and calculate the expectation values of the operators (∆N )2 ,(∆ cos(φ))2 and hsin(φ)i in
state |zi.
3