Quantum Flashcards by David B

What is an operator?

Function which acts upon functions

Can be multiplicative, differential, etc.

Denoted by ^

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What is a linear operator?

Â(c₁f₁+ c₂f₂) = c₁(Âf₁) + c₂(Âf₂)

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Define an eigenfunction

Âf_n = a_nf_n

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

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Define a degenerate eigenfunction

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

Âf₁ = af₁ and Âf₂ = af₂

Â(c₁f₁ + c₂f₂) = a(c₁f₁ + c₂f₂)

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What is the expansion theorem?

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

F = Σ c_nf_n

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What is the 1st postulate of QM?

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)

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What is the 2nd postulate of QM?

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

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

∫ |Ψ(x,t)|² dτ = 1

For all and any t, integral is over all space

From conservation of prob

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

Ψ must be: single-valued, continuous, finite

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What is a KET?

State function independent of representation

|Ψ>

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What is the third postulate of QM?

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

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What is a Hermitian operator?

Operators that are their own Hermitian Conjugate

H = †H

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What is the 4th postulate of QM?

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

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What is the formula of expectation value?

<Â> = ∫ Ψ*ÂΨ dτ = Σ |c_n|²a_n

Where c_n is the coefficient of an eigenfunction

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What is the spread in distribution of measurements given by?

(ΔA)² = <Â²> - <Â>² = Σ P_n a_n²

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What is the position operator?

Observable: position, x

x^ = x

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What is the linear momentum operator?

Observable: p_x

p^_x = -iℏ * δ/δx

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Why are Hermitian operators used for observables?

The eigenvalues are real

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

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What is the kinetic energy QM operator?

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

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What is the time-dependent Schrodinger equation?

ĤΨ(x;t) = iℏ(δ/δt)*Ψ(x;t)

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

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What is a stationary state?

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

What is the commutator of two operators?

[Â,B̂] = ÂB̂ - B̂Â

Evaluated by looking at effect on arbitrary function

What does it mean if operators commute?

[Â,B̂] = 0

Means can measured A and B simultaneously and precisely

What are the properties of commutators?

[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]

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

[Â,B̂] = 0 ÂΨ = aΨ ÂB̂Ψ = B̂ÂΨ = B̂aΨ = aB̂Ψ B̂Ψ is an eigenfn of Â with eigenvalue a, and there is only one eigenfn of Â with value a which is Ψ Simultaneously an eigenfunction of B̂

What is the uncertainty principle?

ΔAΔB ≥ ½ || if commute then = 0, makes sense as no dispersion

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

Ĥ = (1/2m) *p̂_x² =* -(ℏ²/2m)\*d²/dx²

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

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]

What is the de Broglie relation?

λ = h / ρ

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

Ĥ = (1/2m)p̂_x² + V(x) Where V(x) = 0 when inside (0L)

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

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

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

From \<Φ|Φ\> = 1 gives B = Sqrt[2/L]

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

E = (h²/8m) \* [n₁²/L_x² + n₂²/L_y²] If L_x=L_y then states when n₁=n₂ are singly degenerate

What is the potential energy for a classical harmonic oscillator?

V(x) = ½ kx² Force = -V'(x) = -kx

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

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

What is the Hamiltonian for a quantum harmonic oscillator?

Ĥ = p̂_x²/2μ + ½kx² = (-ħ²/2μ)(d²/dx²) + ½kx² as p̂_x = -iħ\*d/dx and x̂ = x

What is the model used for considering molecular vibrations?

Simple Harmonic Oscillator

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

E₀ = ½ħω

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

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

What is the energy levels of simple harmonic oscillators?

E_n = (n+½)ħω and ω = Sqrt[k/μ]

What are the applications of SHO?

Vibrational spec q_vib in stat mechanics Heat capacities of solids

What is the Hamiltonian for a particle on a ring?

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

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

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

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

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]

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

Ψ(Φ) = Sqrt[1/2π] exp(imΦ) where m = 0, +/- 1, +/- 2… E = (m²ħ²/2I) → Energy is doubly degenerate for |m|

What are some applications of particle on a ring?

Free torsional motion Orbital motion in diatomics

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

L̂_z=ħm

How is a particle on a sphere Hamiltonian derived?

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

How can you find if an operator is linear?

Calc both the LHS and RHS to see if equal Â(aΨ₁ + bΨ₂) = a ÂΨ₁ + b ÂΨ₂

How can you find \x²\> of particle in a 1D box?

\x²\> = 2m Inside the box, Ĥ = p̂_x²/2m \<Ĥ\> = \<Ψ_n|Ĥ|Ψ_n\> = E_n = n²h²/8mL² So \x²\> = n²h²/4L²

What is the Hamiltonian of a particle on a sphere?

Ĥ = (-ħ²/2μ) ∇² = L̂²/2I This is in spherical polar coordinates

Where I got up to

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