Summary

Question 1

Wavefuntion

Answer 1

Ψ(x, t)

Question 2

Probability density

Answer 2

P(x, t) = |Ψ(x, t)|²

Question 3

TDSE

Answer 3

ĤΨ = ih ∂Ψ/∂t

Question 4

Hamiltonian operator

Answer 4

Ĥ = p̂²/2m + V̂ = −ħ²/2m ∇² + V

Question 5

Separation of variables

Answer 5

Ψ(x, t) = ψ(x)T(t)

Question 6

TISE

Answer 6

Ĥψ = Eψ

Question 7

Normalization

Answer 7

∫ ψ∗ψ d³x = 1.

Question 8

What are degenerate solutions?

Answer 8

Multiple solutions ψᵢ that have the same energy Eᵢ
.

Question 9

Orthogonality

Answer 9

If Eᵢ ≠ Eⱼ,
∫ ψᵢ ∗ ψⱼ d³x = 0

Question 10

General solution

Answer 10

Ψ(x, t) = Σᵢ Aᵢψᵢ(x)exp(−iEᵢt/ħ)

Question 11

What are observables represented by?

Answer 11

An observable A is represented by an operator Â.

Question 12

Eigenvalue equation of Â

Answer 12

Measurements of A give eigenvalues of Â
ÂΨᵢ = aᵢΨᵢ

Question 13

Hermitian conjugate of Â

Answer 13

† of  is defined such that:
∫ Φ ∗ Â†Ψ d³x = ∫ (ÂΦ) ∗ Ψ d³x

Question 14

Hermitian operator

Answer 14

† = Â

Question 15

2 properties of Hermitian operators

Answer 15

• Only real eigenvalues
• Orthogonal eigenfunctions

Question 16

Expectation value

Answer 16

⟨A⟩ = ∫ Ψ ∗ ÂΨ d³x.

Question 17

Uncertainty

Answer 17

∆A = √⟨²⟩ − ⟨Â⟩² ‘

Question 18

Commutator

Answer 18

[Â, B̂] = ÂB̂ − B̂Â

Question 19

Compatibility

Answer 19

Two measurements “compatible” if [Â, B̂] = 0

Question 20

Uncertainty principle

Answer 20

∆A ∆B ≥|i/2 ⟨[Â, B̂]⟩|

Question 21

Angular momentum operator

Answer 21

L̂ =
|î ĵ k̂|
|x̂ ŷ ẑ |
|p̂ₓ p̂ᵧ p̂ᶻ|

note: î, ĵ and k̂ are unit vectors, not operators!

Question 22

Commutation of angular momentum components.

Answer 22

Angular momentum components are not compatible:
[L̂ₓ, L̂ᵧ] = iħL̂ᶻ, etc

Question 23

Commutation of total angular momentum and one component of angular momentum.

Answer 23

[L̂², L̂ᶻ] = 0
Total angular momentum and any one component are compatible.

Question 24

Eigenvalues of L̂²

Answer 24

l(l + 1)ħ²

Question 25

Eigenvalues of L̂ᶻ

Answer 25

mₗħ

Question 26

Possible values of l and mₗ

Answer 26

l and mₗ are integers, with l ≥ 0 and −l ≤ mₗ ≤ l

Question 27

What are simultaneous eigenfunctions of L̂² and L̂ᶻ?

Answer 27

Spherical harmonics, Yₗₘₗ(θ, ϕ).

Question 28

L̂ᶻ and L̂² in polar coordinates.

Answer 28

L̂ᶻ = −iħ ∂/∂ϕ
L̂² = −ħ² (∂²/∂θ² + cotθ ∂/∂θ + 1/sin²θ ∂²/∂ϕ²)

L̂² only needs to be recognised.

Question 29

Hamiltonian of hydrogen atom

Answer 29

Ĥ = −ħ²/2mₑ (d²/dr² + 2/r d/dr) + L̂²/2mₑr² − e²/4πε₀r

Bracket only needs to be recognised

Question 30

Eigenvalues of hydrogen atom

Answer 30

Eₙ = −Eᴿ/n²
where Eᴿ = mₑe⁴/32π²ε₀²ħ² = 13.6 eV

mₑ term only needs to be recognised, ᴿ should be subscript

Question 31

Why are n, l, and mₗ ‘good’ quantum numbers?

Answer 31

[Ĥ, L̂²] = 0 and [Ĥ, L̂ᶻ] = 0
Measurement of each quantum number doesnt affect the others.

Question 32

Hydrogen wavefunctions
.

Answer 32

ψₙ,ₗ,ₘₗ = const (r/a₀)ˡ exp(−r/na₀) × Qₙ,ₗ (r/a₀) × Pₗ,ₘₗ (θ)exp(imₗϕ)

Question 33

Set of hydrogen wavefunctions with given n and l

Answer 33

Set is called the n“l” orbital.
“l” = s, p, d, f . . . for l = 0, 1, 2, 3, . . ..

Question 34

Eᴿ and a₀ for nucleus of charge Zₑ

R should be subscript

Answer 34

Eᴿ → Z²Eᴿ
a₀ → a₀/Z.

ᴿ should be subscript

Question 35

First order perturbation theory

Answer 35

If Ĥ = Ĥ₀ + V̂ and V̂ is small, use eigenfunctions
of Ĥ₀ with energy shift ⟨V̂⟩

Question 36

Equations of electron spin

Answer 36

⟨Ŝ²⟩ = s(s + 1)ħ²
s = ½
⟨Ŝᶻ⟩ = mₛħ
mₛ = ±½

Question 37

Electron magnetic moment

Answer 37

µ̂ = −µᴮ 1/ħ (L̂ + gŜ)
where µᴮ = eħ/2mₑ and g ≈ 2

Question 38

Stern-Gerlach experiment

Answer 38

Separates a beam of atoms into different components, depending on their µᶻ.

Question 39

Total angular momentum of atom

Answer 39

Ĵ = L̂ + Ŝ

⟨Ĵ²⟩ = j(j + 1)ħ²
⟨Ĵᶻ⟩ = mⱼħ

mⱼ between −j and j
j can be in integer steps between |l − s| and l + s.

⟨L̂²⟩ =l(l + 1)ħ²
⟨L̂ᶻ⟩ = mₗħ

mₗ between −l and l
l is integer 0,1,2…

⟨Ŝ²⟩ = s(s + 1)ħ²
⟨Ŝᶻ⟩ = mₛħ

mₛ = ±½
s = ½

Question 40

Spin-orbit interaction

Answer 40

V̂ ∝ L̂ · Ŝ = ½ (Ĵ² − L̂² − Ŝ²)

Question 41

The four “good” quantum numbers

Answer 41

n - principal quantum number - linked to energy level - 1, 2, 3…

l - azimuthal quantum number - orbital angular momentum - s,p,d,f correspond to 0,1,2,3 - determines total electrons possible in each subshell 2,6,10,14 - possible values 0 to n-1

j - main total angular momentum quantum number - integer steps between |l - s| and l+s (s is spin quantum number)

mⱼ - secondary total angular momentum quantum number - integer steps between -j and j

Question 42

Magnetic energy

Answer 42

If field “weak”, ⟨V̂⟩ = gₗµᴮBmⱼ
where gₗ = “Landé g factor”

if field “strong”, ⟨V̂⟩ = µᴮB(mₗ + gmₛ)

“Weak” and “strong” are relative to size of spin-orbit energy.

gₗ should be L, µᴮ should be subscript B

Question 43

Parity operation

Answer 43

x → −x

Question 44

When are single-photon/radiative transitions allowed?

Answer 44

Only allowed between states of different parity.

Question 45

What does “Fundamental particles are identical” mean?

Answer 45

Fundamental particles are indistinguishable from each other, meaning it is impossible to tell if they have swapped places.

Question 46

Identical particle exchange operation

Answer 46

{x1, s1} ↔ {x2, s2}
Wavefunctions must be symmetric or antisymmetric under exchange.

Question 47

What is symmetry in identical particles?

Answer 47

Particles being indistinguishable means that
|ψ(x₁, x₂)|² = |ψ(x₂, x₁)|²
This is only possible if ψ(x₁, x₂) = ψ(x₂, x₁) OR ψ(x₁, x₂) = -|ψ(x₂, x₁)
for ψ(x₁, x₂) = ψ(x₂, x₁), the particles are symmetric, whereas for ψ(x₁, x₂) = -ψ(x₂, x₁), they are anti-symmetric

Question 48

Examples of particles with symmetric and antisymmetric wavefunctions.

Answer 48

Symmetric: bosons, such as photons
Antisymmetric: fermions, such as electrons

Question 49

Symmetries of electron wavefunctions

Answer 49

Electron wavefunctions have either
Symmetric spin | antisymmetric space
or
Symmetric space | antisymmetric spin

Question 50

Pauli exclusion principle

Answer 50

Two electrons cannot be in an identical state

Question 51

Helium wavefunctions approximation

Answer 51

Helium wavefunctions are approximately the product of two hydrogen wavefunctions plus an upward energy shift due to energy of Coulomb repulsion.

Question 52

Ground state of helium

Answer 52

(1s)²
2 electrons in 1s orbital with antisymmetric electron spins.

Question 53

What is exchange energy?

Answer 53

Exchange energy is a shift in repulsion energy due to fact that electrons are more often closer together/further apart in symmetric/antisymmetric combinations than in independent wavefunctions. It links to parallel and identical electron spin.

Question 54

Effects of filling electron shells on energy.

Answer 54

• Orbitals fill from lowest energy to highest
• Lower orbitals shield nuclear charge from higher orbitals
• Lower l has lower energy because electron spends more time close to origin, inside inner shell and ‘sees’ more of the nucleus.
• Filled orbitals have zero total angular momentum and zero total spin.

Question 55

Hund’s rules

Answer 55

• Maximize S (fill all mₗ ‘slots’ with aligned spins before any with anti-aligned)
• Maximize L (fill slots from high mₗ to low, with L = Σmₗ)
• If orbital is less than half full, minimize J = |L − S|.
• If orbital is more than half full, maximize J = L + S.

Question 56

Spectroscopic term symbol

Answer 56

²ˢ⁺¹“L”ⱼ
e.g. Fe = ⁵D₄

Question 57

Orbital filling order

Answer 57

1s
2s 2p
3s 3p 3d
4s 4p 4d 4f
5s 5p 5d
6s 6p
7s
Order diagonally top right to bottom left
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, . . .