Summary Flashcards
Wavefuntion
Ψ(x, t)
Probability density
P(x, t) = |Ψ(x, t)|²
TDSE
ĤΨ = ih ∂Ψ/∂t
Hamiltonian operator
Ĥ = p̂²/2m + V̂ = −ħ²/2m ∇² + V
Separation of variables
Ψ(x, t) = ψ(x)T(t)
TISE
Ĥψ = Eψ
Normalization
∫ ψ∗ψ d³x = 1.
What are degenerate solutions?
Multiple solutions ψᵢ that have the same energy Eᵢ
.
Orthogonality
If Eᵢ ≠ Eⱼ,
∫ ψᵢ ∗ ψⱼ d³x = 0
General solution
Ψ(x, t) = Σᵢ Aᵢψᵢ(x)exp(−iEᵢt/ħ)
What are observables represented by?
An observable A is represented by an operator Â.
Eigenvalue equation of Â
Measurements of A give eigenvalues of Â
ÂΨᵢ = aᵢΨᵢ
Hermitian conjugate of Â
† of  is defined such that:
∫ Φ ∗ Â†Ψ d³x = ∫ (ÂΦ) ∗ Ψ d³x
Hermitian operator
† = Â
2 properties of Hermitian operators
- Only real eigenvalues
- Orthogonal eigenfunctions
Expectation value
⟨A⟩ = ∫ Ψ ∗ ÂΨ d³x.
Uncertainty
∆A = √⟨²⟩ − ⟨Â⟩² ‘
Commutator
[Â, B̂] = ÂB̂ − B̂Â
Compatibility
Two measurements “compatible” if [Â, B̂] = 0
Uncertainty principle
∆A ∆B ≥|i/2 ⟨[Â, B̂]⟩|
Angular momentum operator
L̂ =
|î ĵ k̂|
|x̂ ŷ ẑ |
|p̂ₓ p̂ᵧ p̂ᶻ|
note: î, ĵ and k̂ are unit vectors, not operators!
Commutation of angular momentum components.
Angular momentum components are not compatible:
[L̂ₓ, L̂ᵧ] = iħL̂ᶻ, etc
Commutation of total angular momentum and one component of angular momentum.
[L̂², L̂ᶻ] = 0
Total angular momentum and any one component are compatible.
Eigenvalues of L̂²
l(l + 1)ħ²
Eigenvalues of L̂ᶻ
mₗħ
Possible values of l and mₗ
l and mₗ are integers, with l ≥ 0 and −l ≤ mₗ ≤ l
What are simultaneous eigenfunctions of L̂² and L̂ᶻ?
Spherical harmonics, Yₗₘₗ(θ, ϕ).
L̂ᶻ and L̂² in polar coordinates.
L̂ᶻ = −iħ ∂/∂ϕ
L̂² = −ħ² (∂²/∂θ² + cotθ ∂/∂θ + 1/sin²θ ∂²/∂ϕ²)
L̂² only needs to be recognised.
Hamiltonian of hydrogen atom
Ĥ = −ħ²/2mₑ (d²/dr² + 2/r d/dr) + L̂²/2mₑr² − e²/4πε₀r
Bracket only needs to be recognised
Eigenvalues of hydrogen atom
Eₙ = −Eᴿ/n²
where Eᴿ = mₑe⁴/32π²ε₀²ħ² = 13.6 eV
mₑ term only needs to be recognised, ᴿ should be subscript
Why are n, l, and mₗ ‘good’ quantum numbers?
[Ĥ, L̂²] = 0 and [Ĥ, L̂ᶻ] = 0
Measurement of each quantum number doesnt affect the others.
Hydrogen wavefunctions
.
ψₙ,ₗ,ₘₗ = const (r/a₀)ˡ exp(−r/na₀) × Qₙ,ₗ (r/a₀) × Pₗ,ₘₗ (θ)exp(imₗϕ)
Set of hydrogen wavefunctions with given n and l
Set is called the n“l” orbital.
“l” = s, p, d, f . . . for l = 0, 1, 2, 3, . . ..
Eᴿ and a₀ for nucleus of charge Zₑ
R should be subscript
Eᴿ → Z²Eᴿ
a₀ → a₀/Z.
ᴿ should be subscript
First order perturbation theory
If Ĥ = Ĥ₀ + V̂ and V̂ is small, use eigenfunctions
of Ĥ₀ with energy shift ⟨V̂⟩
Equations of electron spin
⟨Ŝ²⟩ = s(s + 1)ħ²
s = ½
⟨Ŝᶻ⟩ = mₛħ
mₛ = ±½
Electron magnetic moment
µ̂ = −µᴮ 1/ħ (L̂ + gŜ)
where µᴮ = eħ/2mₑ and g ≈ 2
Stern-Gerlach experiment
Separates a beam of atoms into different components, depending on their µᶻ.
Total angular momentum of atom
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
Spin-orbit interaction
V̂ ∝ L̂ · Ŝ = ½ (Ĵ² − L̂² − Ŝ²)
The four “good” quantum numbers
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
Magnetic energy
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
Parity operation
x → −x
When are single-photon/radiative transitions allowed?
Only allowed between states of different parity.
What does “Fundamental particles are identical” mean?
Fundamental particles are indistinguishable from each other, meaning it is impossible to tell if they have swapped places.
Identical particle exchange operation
{x1, s1} ↔ {x2, s2}
Wavefunctions must be symmetric or antisymmetric under exchange.
What is symmetry in identical particles?
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
Examples of particles with symmetric and antisymmetric wavefunctions.
Symmetric: bosons, such as photons
Antisymmetric: fermions, such as electrons
Symmetries of electron wavefunctions
Electron wavefunctions have either
Symmetric spin | antisymmetric space
or
Symmetric space | antisymmetric spin
Pauli exclusion principle
Two electrons cannot be in an identical state
Helium wavefunctions approximation
Helium wavefunctions are approximately the product of two hydrogen wavefunctions plus an upward energy shift due to energy of Coulomb repulsion.
Ground state of helium
(1s)²
2 electrons in 1s orbital with antisymmetric electron spins.
What is exchange energy?
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.
Effects of filling electron shells on energy.
- 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.
Hund’s rules
- 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.
Spectroscopic term symbol
²ˢ⁺¹“L”ⱼ
e.g. Fe = ⁵D₄
Orbital filling order
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, . . .
