Equations

Question 1

creation operator

Answer 1

a† = 1/sqrt(2ℏω) (ωq-ip)

Question 1

annihilation operator

Answer 1

a = 1/sqrt(2ℏω) (ωq+ip)

Question 2

commutator relation between the creation and annihilation operator

Answer 2

[a, a†] = aa† - a†a = 1

Question 3

when is something hermitian?

Answer 3

when q = q†

Question 4

useful commutator relation

Answer 4

[A,B] = -[B,A]

Question 5

ΔX

Answer 5

ΔX = sqrt(<x^2> - <x>^2)</x>

Question 6

operator identity

Answer 6

[A, [A,B]] = [B, [A,B]] = 0

Question 7

beamsplitter relations

Answer 7

set up as a matrix system

(a3 a4)^T = (t r r t)^T (a1 a2)^T

Question 8

how to determine the inverse of a 2 x 2 matrix

Answer 8

(a b c d)^-1 = 1/ad-bc (d -b -c a)

Question 9

relationship between transmission and reflection

Answer 9

|t|^2 + |r|^2 = 1

Question 10

number state

Answer 10

n = a†a

Question 11

coherent states

Answer 11

a|α> = α|α>

<α|a(dagger) = <α|α*

Question 12

normal ordering

Answer 12

all a† to the left

Question 13

n(hat)|n>

Answer 13

= n|n>

Question 14

<m|n>

Answer 14

δ_mn

Question 15

δmn =

Answer 15

{1 if m = n
{0 if m ≠ n

Question 16

Σm Σn δmn Am Bn =

Answer 16

Σm Am Bm

Question 17

binomial theorem

Answer 17

(x + y)^n = (n Σ k=0) (n k)^T (x)^k (y)^(n-k)

where (n k)^T = n!/k!(n-k)!

Question 18

|n> =

Answer 18

1/sqrt(n!) (a†)^n |0 >

Question 19

n!/sqrt(n!)

Answer 19

= sqrt(n!)

Question 20

beamsplitter something?

Answer 20

|a1|^2 + |a2|^2 = |a3|^2 + |a4|^2

Question 21

what is the expectation value of the number operator in a coherent state?

Answer 21

the coherent state modulus squared

i.e.

|α|^2

Question 22

completeness relation

Answer 22

Σi |i><i| = 𝕀 = 1 in the discrete case

∫ dx|x><x| = 𝕀

Question 23

e^x

Answer 23

Σ x^n/n!

Question 24

Σ exp(-nβ)

Answer 24

= 1/(1-exp(-β))

Question 25

a†|n>

Answer 25

sqrt(n+1)|n+1>

Question 26

a|n>

Answer 26

sqrt(n)|n-1>

Question 27

|α>

Answer 27

exp(-|α|^2/2) Σ α^n/sqrt(n!) | n>

Question 28

D0(α)

Answer 28

exp(-|α|^2/2 + αa†)

Question 29

D(α)

Answer 29

exp(αa† - α*a)

Question 30

density operator

Answer 30

p = Σ pn|n>

Question 31

50:50 beamsplitter

Answer 31

r = i/sqrt(2) and t = 1/sqrt(2)

Question 32

Wigner function

Answer 32

W(q,p) on the formula sheet

Question 33

Hamiltonian

Answer 33

H = hω(a†a + 1/2)

Question 34

Energy levels of the quantum harmonic oscillator

Answer 34

En = hω(n+1/2)

Question 35

ladder operators and quadrature operators

Answer 35

a = X + iY a† = X - iY

Question 36

χ(r,t)

Answer 36

ωk t - k.r - π/2

Question 37

commutator of a single mode quadrature operators

Answer 37

[X,Y] = i/2

Question 38

uncertainty relation of quadrature operators

Answer 38

ΔXΔY > 1/4

Question 39

the expectation value of an arbitrary operator O when the system is in a mixed state is described by the density operator

Answer 39

p = Tr(pO)

Question 40

trace operator

Answer 40

Tr(O) = Σi <ψi|O|ψi>

Question 41

Partition function

Answer 41

Z = Σn exp(-En/kbT)

Question 43

Density operator for thermal state of a multi mode electromagnetic field

Answer 43

pth = 1/Z exp(-H/kbT) Z = Tr[exp(-H/kbT)]

Question 45

Density operator for thermal state of a single mode electromagnetic field

Answer 45

pth,1 = (1-exp(-β)) Σ exp(-nβ) |n>