Quantum

Question 1

What is an operator?

Answer 1

Function which acts upon functions

Can be multiplicative, differential, etc.

Denoted by ^

Question 2

What is a linear operator?

Answer 2

Â(c₁f₁ + c₂f₂) = c₁(Âf₁) + c₂(Âf₂)

Question 3

Define an eigenfunction

Answer 3

Âf_n = a_nf_n

f_n is an eigenfunction if has form above with a_n being the eigenvalue (is a constant)

Question 4

Define a degenerate eigenfunction

Answer 4

Two eigenfunctions (f₁ and f₂) which possess a common eigenvalue

Âf₁ = af₁ and Âf₂ = af₂

Â(c₁f₁ + c₂f₂) = a(c₁f₁ + c₂f₂)

Question 5

What is the expansion theorem?

Answer 5

Every function can be expanded in terms of eigenfunctions of an operator

F = Σ c_nf_n

Question 6

What is the 1st postulate of QM?

Answer 6

State of a system of N particles is fully described by a function called the wavefunction

Has the form: Ψ(r₁,r₂,…,r_n;t)

Question 7

What is the 2nd postulate of QM?

Answer 7

Prob that a system in state Ψ will be found in infinitesimal volume element (dτ)

Prob = Ψ*(x,t)Ψ(x,t)dx = |Ψ(x,t)|² dx

This is a statistical probability

Question 8

What is the normalisation condition for the born’s probability?

Answer 8

∫ |Ψ(x,t)|² dτ = 1

For all and any t, integral is over all space

From conservation of prob

Question 9

What is some requirements of the wavefn which comes from the born probability?

Answer 9

Ψ must be: single-valued, continuous, finite

Question 10

What is a KET?

Answer 10

State function independent of representation

|Ψ>

Question 11

What is the third postulate of QM?

Answer 11

Observables represented by operators - for every classical observable A there is a corresponding QM operator which is linear and Hermitian

Question 12

What is a Hermitian operator?

Answer 12

Operators that are their own Hermitian Conjugate

H = †H

Question 13

What is the 4th postulate of QM?

Answer 13

Single measurement of an observable A yields a single result, which is one of the eigenvalues (a_n) of Â

Mean value of A obtained from many measured is equal to expectation value of corresponding operator

Question 14

What is the formula of expectation value?

Answer 14

<Â> = ∫ Ψ*ÂΨ dτ = Σ |c_n|²a_n

Where c_n is the coefficient of an eigenfunction

Question 15

What is the spread in distribution of measurements given by?

Answer 15

(ΔA)² = <²> - <Â>² = Σ P_n a_n²

Question 16

What is the position operator?

Answer 16

Observable: position, x

x^ = x

Question 17

What is the linear momentum operator?

Answer 17

Observable: p_x

p^_x = -iℏ * δ/δx

Question 18

Why are Hermitian operators used for observables?

Answer 18

The eigenvalues are real

Eigenfunctions corresponding to different eigenvalues of Hermitian operators are orthogonal, i.e. m|f_n> = 0

Question 19

What is the kinetic energy QM operator?

Answer 19

T^ = ½ (p̂_xp̂_{x +} p̂_yp̂_y + p̂_zp̂_z) = -ℏ²/2m * (δ²/δx² + δ²/δy² + δ²/δz²)

Question 20

What is the time-dependent Schrodinger equation?

Answer 20

ĤΨ(x;t) = iℏ(δ/δt)*Ψ(x;t)

This is due to iℏ(δ/δt) being defined as a total energy operator

Question 21

What is a stationary state?

Answer 21

This is when no measurable property of the system evolves or changes with time

Question 22

What is the commutator of two operators?

Answer 22

[Â,B̂] = ÂB̂ - B̂Â

Evaluated by looking at effect on arbitrary function

Question 23

What does it mean if operators commute?

Answer 23

[Â,B̂] = 0

Means can measured A and B simultaneously and precisely

Question 24

What are the properties of commutators?

Answer 24

[B̂,Â] = - [Â,B̂]

[Â,cB̂] = c[Â,B̂] where c is a pure number

[Â,B̂+Ĉ] = [Â,B̂] + [Â,Ĉ]

[Â,B̂Ĉ] = [Â,B̂]Ĉ + B̂[Â,Ĉ]

Question 25

What occurs when a non-degen wavefn of  is acted on by two operators which commutes?

Answer 25

[Â,B̂] = 0

ÂΨ = aΨ

ÂB̂Ψ = B̂ÂΨ = B̂aΨ = aB̂Ψ

B̂Ψ is an eigenfn of  with eigenvalue a, and there is only one eigenfn of  with value a which is Ψ

Simultaneously an eigenfunction of B̂

Question 26

What is the uncertainty principle?

Answer 26

ΔAΔB ≥ ½ ||

if commute then = 0, makes sense as no dispersion

Question 27

What is the Hamiltonian operator for a 1D free particle?

Answer 27

Ĥ = (1/2m) p̂_x² = -(ℏ²/2m)*d²/dx²

Question 28

What is the physical intepretation of sign p̂_x = -(iℏ)δ/δx ?

Answer 28

if p̂_xΦ = +ℏkΦ then particle moving in a direction with momentum +ℏk

if p̂_xΦ = -ℏkΦ then particle moving in opposite direction with momentum -ℏk

where mag. of linear momentum, ρ = ℏk = Sqrt[2mE]

Question 29

What is the de Broglie relation?

Answer 29

λ = h / ρ

Question 30

What is the Hamiltonian for a particle in a 1D box?

Answer 30

Ĥ = (1/2m)p̂_x² + V(x)

Where V(x) = 0 when inside (0L)

Question 31

What is the solution to the energy of a particle in a 1D box?

Answer 31

Φ(x) = A*cos(x p/ℏ) + Bsin(x p/ℏ)

Boundary cond. → Φ(0) = 0 = A and Φ(L) = 0 = Bsin(x p/ℏ)

so Φ(x) = Bsin(x p/ℏ)

Φ(L) = Bsin(L p/ℏ) → L p/ℏ = nπ

p = nh/2L → Quantised magnitude of momentum , and as p = Sqrt[2mE]

E = n²h²/8mL²

Question 32

How can the normalisation constant in a particle in a box be found?

Answer 32

From <Φ|Φ> = 1 gives B = Sqrt[2/L]

Question 33

What is the energy of a particle in a 2D box?

Answer 33

E = (h²/8m) * [n₁²/L_x² + n₂²/L_y²]

If L_x=L_y then states when n₁=n₂ are singly degenerate

Question 34

What is the potential energy for a classical harmonic oscillator?

Answer 34

V(x) = ½ kx²

Force = -V’(x) = -kx

Question 35

What is the energy of a harmonic oscillator in classical mechanics?

Answer 35

E = PE + KE = ½kx² + ½μv² = ½kx² + p_x²/2μ

Where p_x = μv

Question 36

What is the Hamiltonian for a quantum harmonic oscillator?

Answer 36

Ĥ = p̂_x²/2μ + ½kx² = (-ħ²/2μ)(d²/dx²) + ½kx²

as p̂_x = -iħ*d/dx and x̂ = x

Question 37

What is the model used for considering molecular vibrations?

Answer 37

Simple Harmonic Oscillator

Question 38

What is the zero-point energy of a simple harmonic oscillator?

Answer 38

E₀ = ½ħω

Question 39

What do the solutions to the QM simple harmonic oscillator and what does it suggest?

Answer 39

Suggests: Zero-point energy and particle can exist in classically forbidden region (tunnelling)

Question 40

What is the energy levels of simple harmonic oscillators?

Answer 40

E_n = (n+½)ħω

and ω = Sqrt[k/μ]

Question 41

What are the applications of SHO?

Answer 41

Vibrational spec

q_vib in stat mechanics

Heat capacities of solids

Question 42

What is the Hamiltonian for a particle on a ring?

Answer 42

Ĥ = p̂_x²/2μ + p̂_y²/2μ = -(ħ²/2μ) (δ²/δx² + δ²/δy²)

then in polar coordinates (use this one):

Ĥ = -(ħ²/2μr²) δ²/δΦ² =-(ħ²/2I) δ²/δΦ²

Question 43

What is the z-component of angular momentum operator and how is it used in particle on a ring?

Answer 43

L̂_z = -iħ δ/δΦ

used to sub into hamiltonian: Ĥ = L̂_z²/2I

[Ĥ,L̂_z] = 0 so eigenfunctions of one are of the other

Question 44

What boundary condition can be used for a particle on a ring?

Answer 44

Ψ(Φ+2π) = Ψ(Φ)exp[im2π]

where m = 0, +/- 1, +/- 2…

Where Ψ(Φ) = N*exp[imΦ]

Gives normalisation constant, N = Sqrt[1/2π]

Question 45

What is the wavefunction and energy for a particle on a ring?

Answer 45

Ψ(Φ) = Sqrt[1/2π] exp(imΦ) where m = 0, +/- 1, +/- 2…

E = (m²ħ²/2I) → Energy is doubly degenerate for |m|

Question 46

What are some applications of particle on a ring?

Answer 46

Free torsional motion

Orbital motion in diatomics

Question 47

What is the momentum on the z axis in a particle on a ring?

Answer 47

L̂_z=ħm

Question 48

How is a particle on a sphere Hamiltonian derived?

Answer 48

Start with a particle on a ring:

fix r → transform to spherical polar coord

use separation of variables to separate θ and Φ parts in the Hamiltonian

Question 49

How can you find if an operator is linear?

Answer 49

Calc both the LHS and RHS to see if equal

Â(aΨ₁ + bΨ₂) = a ÂΨ₁ + b ÂΨ₂

Question 50

How can you find <p̂_x²> of particle in a 1D box?

Answer 50

<p̂_x²> = 2m

Inside the box, Ĥ = p̂_x²/2m

<Ĥ> = <Ψ_n|Ĥ|Ψ_n> = E_n = n²h²/8mL²

So <p̂_x²> = n²h²/4L²

Question 51

What is the Hamiltonian of a particle on a sphere?

Answer 51

Ĥ = (-ħ²/2μ) ∇² = L̂²/2I

This is in spherical polar coordinates

Question 52

Where I got up to

Answer 52

Notes Lectures 5-12 MW → Separation of variables, p 13