Comments
Description
Transcript
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