Quantum Flashcards
What is an operator?
Function which acts upon functions
Can be multiplicative, differential, etc.
Denoted by ^
What is a linear operator?
Â(c1f1 + c2f2) = c1(Âf1) + c2(Âf2)
Define an eigenfunction
Âfn = anfn
fn is an eigenfunction if has form above with an being the eigenvalue (is a constant)
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)
What is the expansion theorem?
Every function can be expanded in terms of eigenfunctions of an operator
F = Σ cnfn
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,…,rn;t)
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
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
What is some requirements of the wavefn which comes from the born probability?
Ψ must be: single-valued, continuous, finite
What is a KET?
State function independent of representation
|Ψ>
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
What is a Hermitian operator?
Operators that are their own Hermitian Conjugate
H = †H
What is the 4th postulate of QM?
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
What is the formula of expectation value?
<Â> = ∫ Ψ*ÂΨ dτ = Σ |cn|2an
Where cn is the coefficient of an eigenfunction
What is the spread in distribution of measurements given by?
(ΔA)2 = <Â2> - <Â>2 = Σ Pn an2
What is the position operator?
Observable: position, x
x^ = x
What is the linear momentum operator?
Observable: px
p^x = -iℏ * δ/δx
Why are Hermitian operators used for observables?
The eigenvalues are real
Eigenfunctions corresponding to different eigenvalues of Hermitian operators are orthogonal, i.e. m|fn> = 0
What is the kinetic energy QM operator?
T^ = ½ (p̂xp̂x + p̂yp̂y + p̂zp̂z) = -ℏ2/2m * (δ2/δx2 + δ2/δy2 + δ2/δz2)
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
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  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
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) p̂x2 = -(ℏ2/2m)*d2/dx2
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]
What is the de Broglie relation?
λ = h / ρ
What is the Hamiltonian for a particle in a 1D box?
Ĥ = (1/2m)p̂x2 + 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/Lx2 + n22/Ly2]
If Lx=Ly 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 + px2/2μ
Where px = μv
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
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?
En = (n+½)ħω
and ω = Sqrt[k/μ]
What are the applications of SHO?
Vibrational spec
qvib in stat mechanics
Heat capacities of solids
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
What is the z-component of angular momentum operator and how is it used in particle on a ring?
L̂z = -iħ δ/δΦ
used to sub into hamiltonian: Ĥ = L̂z2/2I
[Ĥ,L̂z] = 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?
L̂z=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 <p̂x2> of particle in a 1D box?
<p̂x2> = 2m
Inside the box, Ĥ = p̂x2/2m
<Ĥ> = <Ψn|Ĥ|Ψn> = En = n2h2/8mL2
So <p̂x2> = 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
