Quantum Flashcards

1
Q

What is an operator?

A

Function which acts upon functions

Can be multiplicative, differential, etc.

Denoted by ^

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2
Q

What is a linear operator?

A

Â(c1f1 + c2f2) = c1(Âf1) + c2(Âf2)

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3
Q

Define an eigenfunction

A

Âfn = anfn

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

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4
Q

Define a degenerate eigenfunction

A

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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5
Q

What is the expansion theorem?

A

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

F = Σ cnfn

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6
Q

What is the 1st postulate of QM?

A

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

Has the form: Ψ(r1,r2,…,rn;t)

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7
Q

What is the 2nd postulate of QM?

A

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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8
Q

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

A

∫ |Ψ(x,t)|2 dτ = 1

For all and any t, integral is over all space

From conservation of prob

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9
Q

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

A

Ψ must be: single-valued, continuous, finite

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10
Q

What is a KET?

A

State function independent of representation

|Ψ>

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11
Q

What is the third postulate of QM?

A

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

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12
Q

What is a Hermitian operator?

A

Operators that are their own Hermitian Conjugate

H = H

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13
Q

What is the 4th postulate of QM?

A

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

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

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14
Q

What is the formula of expectation value?

A

<Â> = ∫ Ψ*ÂΨ dτ = Σ |cn|2an

Where cn is the coefficient of an eigenfunction

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15
Q

What is the spread in distribution of measurements given by?

A

(ΔA)2 = <Â2> - <Â>2 = Σ Pn an2

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16
Q

What is the position operator?

A

Observable: position, x

x^ = x

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17
Q

What is the linear momentum operator?

A

Observable: px

p^x = -iℏ * δ/δx

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18
Q

Why are Hermitian operators used for observables?

A

The eigenvalues are real

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

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19
Q

What is the kinetic energy QM operator?

A

T^ = ½ (p̂xx + yy + p̂zz) = -ℏ2/2m * (δ2/δx2 + δ2/δy2 + δ2/δz2)

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20
Q

What is the time-dependent Schrodinger equation?

A

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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21
Q

What is a stationary state?

A

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

22
Q

What is the commutator of two operators?

A

[Â,B̂] = ÂB̂ - B̂Â

Evaluated by looking at effect on arbitrary function

23
Q

What does it mean if operators commute?

A

[Â,B̂] = 0

Means can measured A and B simultaneously and precisely

24
Q

What are the properties of commutators?

A

[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]

25
What occurs when a non-degen wavefn of  is acted on by two operators which commutes?
[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
26
What is the uncertainty principle?
ΔAΔB ≥ ½ || if commute then = 0, makes sense as no dispersion
27
What is the Hamiltonian operator for a 1D free particle?
Ĥ = (1/2m) *p̂x2 =* -(ℏ2/2m)\*d2/dx2
28
What is the physical intepretation of sign p̂x = -(iℏ)δ/δx ?
if p̂xΦ = +ℏkΦ then particle moving in a direction with momentum +ℏk if p̂xΦ = -ℏkΦ then particle moving in opposite direction with momentum -ℏk where mag. of linear momentum, ρ = ℏk = Sqrt[2mE]
29
What is the de Broglie relation?
λ = h / ρ
30
What is the Hamiltonian for a particle in a 1D box?
Ĥ = (1/2m)p̂x2 + V(x) Where V(x) = 0 when inside (0L)
31
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
32
How can the normalisation constant in a particle in a box be found?
From \<Φ|Φ\> = 1 gives B = Sqrt[2/L]
33
What is the energy of a particle in a 2D box?
E = (h2/8m) \* [n12/Lx2 + n22/Ly2] If Lx=Ly then states when n1=n2 are singly degenerate
34
What is the potential energy for a classical harmonic oscillator?
V(x) = ½ kx2 Force = -V'(x) = -kx
35
What is the energy of a harmonic oscillator in classical mechanics?
E = PE + KE = ½kx2 + ½μv2 = ½kx2 + px2/2μ Where px = μv
36
What is the Hamiltonian for a quantum harmonic oscillator?
Ĥ = p̂x2/2μ + ½kx2 = (-ħ2/2μ)(d2/dx2) + ½kx2 as p̂x = -iħ\*d/dx and x̂ = x
37
What is the model used for considering molecular vibrations?
Simple Harmonic Oscillator
38
What is the zero-point energy of a simple harmonic oscillator?
E0 = ½ħω
39
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)
40
What is the energy levels of simple harmonic oscillators?
En = (n+½)ħω and ω = Sqrt[k/μ]
41
What are the applications of SHO?
Vibrational spec qvib in stat mechanics Heat capacities of solids
42
What is the Hamiltonian for a particle on a ring?
Ĥ = p̂x2/2μ + p̂y2/2μ = -(ħ2/2μ) (δ2/δx2 + δ2/δy2) then in polar coordinates (use this one): Ĥ = -(ħ2/2μr2) δ2/δΦ2 =-(ħ2/2I) δ2/δΦ2
43
What is the z-component of angular momentum operator and how is it used in particle on a ring?
z = -iħ δ/δΦ used to sub into hamiltonian: Ĥ = L̂z2/2I [Ĥ,L̂z] = 0 so eigenfunctions of one are of the other
44
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]
45
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|
46
What are some applications of particle on a ring?
Free torsional motion Orbital motion in diatomics
47
What is the momentum on the z axis in a particle on a ring?
z=ħm
48
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
49
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
50
How can you find \x2\> of particle in a 1D box?
\x2\> = 2m Inside the box, Ĥ = p̂x2/2m \<Ĥ\> = \<Ψn|Ĥ|Ψn\> = En = n2h2/8mL2 So \x2\> = n2h2/4L2
51
What is the Hamiltonian of a particle on a sphere?
Ĥ = (-ħ2/2μ) ∇2 = L̂2/2I This is in spherical polar coordinates
52
Where I got up to
Notes Lectures 5-12 MW → Separation of variables, p 13