Formula sheet for quantum exam (DRAFT)

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General—operators, etc

〈Q〉 =∑

n

qn|cn|2 [AB,C] = A[B,C] + [A,C]B

Harmonic oscillator

a± ≡ 1√2hmω

(∓ip+mωx) [a−, a+] = 1 H = hω(a+a− +12)

a+ψn =√n+ 1 ψn+1 a−ψn =

√n ψn−1 ψ0(x) =

(mωπh

)1/4

exp(−mω2h

x2)

Boundary conditions in 1DIf V (x) = αδ(x− a) + (bounded), then the discontinuity of the derivative of the eigenfunction at x = a is given by

∆dψ(x)dx

∣∣∣∣a

=2mh2 αψ(a)

Angular Momentum

L± ≡ Lx ± iLy L2Y ml = l(l + 1)h2Y m

l LzYml = mhY m

l L±Yml = h

√l(l + 1)−m(m± 1)Y m±1

l

Perturbation theory

E1n = 〈ψ0

n|H ′ψ0n〉 E2

n =∑m6=n

|〈ψ0m|H ′ψ0

n〉|2

E0n − E0

m

ca = − i

hH ′

abe−iω0tcb cb = − i

hH ′

baeiω0tca

Mathematics

exp(x) = 1 + x+12!x2 +

13!x3 + . . .

sin(x) = x− 13!x3 +

15!x5 − . . .

cos(x) = 1− 12!x2 +

14!x4 − . . .

(1 + x)α = 1 + αx+α(α− 1)(1)(2)

x2 + . . .

f(x) = f(a) + f ′(a)(x− a) +12!

(x− a)2 + . . .

∫ ∞

0

exp(−ax2)dx =√π

21a1/2∫ ∞

0

x exp(−ax2)dx =12

1a∫ ∞

0

x2 exp(−ax2)dx =√π

41a3/2∫ ∞

0

x3 exp(−ax2)dx =12

1a2∫ ∞

0

x4 exp(−ax2)dx =3√π

81a5/2

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