formula sheet for quantum exam (draft)dirac.ruc.dk/~nbailey/qm2008/formulasheet.pdf · formula...
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Formula sheet for quantum exam (DRAFT)
1
2
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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