Formulae

Question 1

the overlap of the same state

Answer 1

<n|n> = 1

Question 2

commutator

Answer 2

[A,B] = [AB-BA]

Question 3

something is minimised when

Answer 3

the derivative is set equal to zero i.e. d/dx = 0

Question 4

reduced mass

Answer 4

µ = (me mN)/(me+mN) = (m1 m2)/(m1+m2)

Question 5

spherical harmonics are correctly normalised if

Answer 5

(2π ∫ 0) (π ∫ 0) |(Y^m l)|^2 sinθdθdφ

Remember to square everything in the wave function

Question 6

a wave function is normalised in general if

Answer 6

( ∞ ∫ 0) |Ψ|^2 dr^3 = 1

Question 7

expectation value of <r^3>

Answer 7

calculate ∫dr^3 |Ψ|^2 r^3 as before with extra expectation value multiplied in

Question 8

Rydberg equation

Answer 8

E = Ry (1/n2^2 - 1/n1^2)

Question 9

Planck’s equation

Answer 9

E = hf = hλ/c

Question 10

Derive the angular momentum operators Lx, Ly and Lz

Answer 10

L = r x p

p = (-iℏ∇)

and r = (x,y,z)

take the cross product

Question 11

lowering operator for L

Answer 11

L- = Lx - iLy

Question 12

the conjugate of |x>

Answer 12

<x|

Question 13

total angular momentum quantum number

Answer 13

j = l ± 1/2

Question 14

how is mj related to ms and ml

Answer 14

mj = ml + ms

Question 15

how is J related to L and S

Answer 15

J = L + S

Question 16

first order pertubation theory

Answer 16

|Ψn(λ)> = |Ψn(0)> + λ|Ψn(1)> + O(λ^2)

Question 17

to show something is orthonormal

Answer 17

take the conjugate

Question 18

hydrogen atom wavefunction

Answer 18

Ψnlm on formula sheet

Question 19

virial theorem

Answer 19

<V> = -2<T>
</T></V>

Question 20

Write the Hamiltonian in the form of its kinetic and potential energy

Answer 20

<H> = <T> + <V>
</V></T></H>

Question 21

how are the relativstic corrections and the spin-orbit coupling related

Answer 21

En^(0) = E^(1) rel + E^(1) SO

on the formula sheet = En (Zα)^2/n^2 …

Question 22

in the stark effect z can be written as

Answer 22

z = rcosθ

Question 23

energy shift in the Paschen-back regime

Answer 23

strong magnetic field

∆E = ωlℏ(m+2ms) on formula sheet

Question 24

how is m related to l

Answer 24

-l ≤ m ≤ l

Question 25

what does >JJ mean

Answer 25

j,mj>

Question 26

what does >LS mean

Answer 26

ml,ms>

Question 27

how is >JJ related to >LS

Answer 27

|jmax,jmax>JJ = l,s>LS

Question 28

how does n relate to l

Answer 28

n = 1,2,3,... l = 0,1,2,...

Question 29

the total energy of two electrons to first order pertubation

Answer 29

Etot = E(0) + E(1)

Question 30

binding energy

Answer 30

En ~ - Ry Z^2/n^2

Question 31

transition energy

Answer 31

E(gamma) = Ry(Z^2/n1^2 - Z^2/n2^2)

Question 32

labelling of shells and subshells

Answer 32

K, L, M, N n = 1,2,3,4 l = 0,1,2,3 s, p, d, f n = 1,2,3,4 l = 0,1,2,3

Question 33

[p,x] =

Answer 33

iℏ

Question 34

quantum harmonic oscillator ladder operators

Answer 34

a-|n> = √n|n-1> a+|n> = √n+1|n+1>

Question 35

harmonic oscillator restoring force

Answer 35

F = -kx = - mω^2x

Question 36

force constant

Answer 36

k = mω^2

Question 37

what is the first order perturbation

Answer 37

H(1)

Question 38

what is the energy of the first order perturbation

Answer 38

E(1) =

Question 39

rotational energy

Answer 39

E = 1/2ℏω

Question 40

reduced mass for hydrogen

Answer 40

µ = 1/2 mp

Question 41

Operator L^2 and S^2 and J^2

Answer 41

J = h(bar)^2 j(j+1) similarly for L and S

Question 42

Zeroth order perturbation

Answer 42

H(0)|n> = En(0)|n>

Answer 43

δnm

Question 44

Standard deviation

Answer 44

Δr = ( -^2)^(1/2)

Question 45

Stark energy shift

Answer 45

ΔEstark = eEz.z Hstark = εE.z eε ∫ z dr^3 |psi|^2

Question 46

Diagonalisation of the matrix of eigenvalues

Answer 46

det|H(1) - λI| gives the energy splitting

Question 47

Applying the lowering operator to the ground state

Answer 47

a-|n0> = 0

Question 48

K alpha

Answer 48

n = 2 to n= 1

Question 49

K beta

Answer 49

n = 3 to n = 1

Question 50

Energy levels for harmonic oscillators.

Answer 50

En = ℏω(n+1/2)

Question 51

The number of spin states for para spin states

Answer 51

N = (s+1)(2s+1)

Question 52

The number of spin states for ortho spin states

Answer 52

N = s(2s+1)

Question 53

In a weak magnetic field

Answer 53

(L . 2S) . B is small

Question 54

Re write Lz + 2Sz in terms of Jz and Sz

Answer 54

Lz + Sz + Sz => Jz + Sz

Question 55

x(hat)

Answer 55

x(hat) = i (ℏ/2mω)^1/2 (A- - A+)

Question 56

p(hat)

Answer 56

p(hat) = (mℏω/2)^1/2 (a+ + a-)

Question 57

Screening factors

Answer 57

Z1 ~ (Z-a) and Z2 ~ (Z-b)

Question 58

Parity

Answer 58

Pr = r Pθ = π - θ PΦ = Φ + π

Question 59

the orbital magnetic moment

Answer 59

mL = -e/2me L

Question 60

L x L

Answer 60

iℏL

Question 61

S x S

Answer 61

iℏS

Question 62

spin magnetic moment

Answer 62

ms = -eg/2me S

Question 63

nuclear magneton

Answer 63

μN = eℏ/2mN

Question 64

{A,B}

Answer 64

AB + BA

Question 65

energy of a multielectron states

Answer 65

Ex(0) = -Z^2 Ry (1/n1^2 - 1/n2^2)

Question 66

state labelling the the L-S coupling scheme (diatomic molecules)

Answer 66

2S+1 XJ where X = S,P,D,F

Question 67

S1 . S2

Answer 67

S(1,+)S(2,-) + S(1,-)S(2,+) + 2S(1,z)S(2,z)

Question 68

L . S

Answer 68

L+S- + L-S+ + 2LzSz

Question 69

[Lz,z]

Answer 69

=0

Question 70

[Lz,L±]

Answer 70

±ħL±

Question 71

[L^2,L ±]

Answer 71

=0

Question 72

[L,S]

Answer 72

=0

Question 73

A probability distribution is maximised

Answer 73

When x tends to infinity

Question 74

Energy shift to First order perturbation of the stark effect

Answer 74

En < psi | eεz | psi > = eε ∫ dr^3 z |psi|^2

Question 75

Probability density

Answer 75

|Rnl|^2 |Yml|^2 r^2 sin theta d theta d phi dr

Question 76

Property of factorials

Answer 76

n! = n x (n-1)!

Question 77

|jmax, -jmax> JJ

Answer 77

= | -l, -s>LS

Question 78

|jmax,jmax>JJ

Answer 78

|l,s>