Particles, Quanta & Fields Flashcards
Give examples of an EM wave acting as a particle.
- Compton scattering
- Photoelectric effect
- Black body radiation
Give an example of a particle acting as a wave.
Double slit experiment with electrons.
What is the Heisenberg Uncertainty Principle?
∆x ∆p ≥ ħ / 2
Where:
∆x is the uncertainty in position
∆p is the uncertainty in momentum
What is the 1D TDSE?
iħ 𝜕Ψ / 𝜕t = ( - ħ^2/ 2m ) (𝜕^2Ψ(x, t) / 𝜕x^2) + V(x)Ψ(x, t)
What is the 1D TISE?
(- ħ^2/2m) (d^2Ψ(x) / dx^2) + V(x)Ψ(x) = EΨ(x)
What is the variational principle?
The variational principle is a way to approximate the ground state.
Give the equation for the Hamiltonian.
^H = - (ħ^2 / 2m) (∂^2 / ∂x^2) + V(x)
What is the definition of the Hamiltonian?
The sum of the kinetic and potential energies of all particles associated with the system.
What is i^2?
i^2 = -1
If z = x + iy what is the complex conjugate of z?
z* = x - iy
What is normalisation?
∫ (∞, -∞) |Ψ(x)^2| dx = 1
What is the equation for wavenumber, k?
k = 2π / λ = 2πf / c
State what it means for a wavefunction |φ〉to be an eigenfunction of an operator ˆA.
|φ〉 is an eigenfunction of an operator ˆA if:
ˆA|φ = λ |φ〉
where λ is the eigenvalue corresponding to the eigenfunction |φ〉.
What is the dirac notation?
〈φ|ψ〉 =∫ (∞, −∞) φ∗ψ dx
How do you find the complex conjugate of something?
You find the complex conjugate by changing the sign of the imaginary part of the complex number.
What is the operator form of the Schrodinger equation?
^H ψ = E ψ
What is the momentum operator?
^ρ = (ħ / i) (𝜕 / 𝜕x)
In quantum mechanics, what are physical quantities represented by?
Operators
What is the 3D TDSE?
iħ 𝜕Ψ / 𝜕t = ( - ħ^2/ 2m ) ((𝜕^2 / 𝜕x^2) + (𝜕^2 / 𝜕y^2) + (𝜕^2 / 𝜕z^2))Ψ + V(x, y, z)Ψ
What is the 3D TISE?
(- ħ^2/2m) ((𝜕^2 / 𝜕x^2) + (𝜕^2 / 𝜕y^2) + (𝜕^2 / 𝜕z^2))ψ + V(x, y, z)ψ = Eψ
What is the 3D Hamiltonian?
^H = - (ħ^2 / 2m) ((𝜕^2 / 𝜕x^2) + (𝜕^2 / 𝜕y^2) + (𝜕^2 / 𝜕z^2))+ V(x, y, z)
What is the expectation value definition?
The expectation value is the probabilistic expected value of the result (measurement) of an experiment.
What is the expectation value?
⟨A⟩ = ⟨ψ|A|ψ⟩
An operator is called Hermitian if…
⟨𝜙 ∨ ^A 𝜓⟩ = ⟨^A 𝜙 ∨ 𝜓⟩
Of which operators are the spherical harmonics Y(lm) eigenfunctions? What are the corresponding eigenvalues?
^(hat)L^2 whose eigenvalue is l (l + 1) ħ^2
^(hat)L(z) whose eigenvalue is mħ
In the context of Perturbation Theory, state the formula for the first order correction to the energy in terms of the perturbing Hamiltonian ^H
E(n)^(1) = (⟨ 𝜓(n)^(0) | ^H^(1) | 𝜓(n)^(0) ⟩) / (⟨ 𝜓(n)^(0) | 𝜓(n)^(0) ⟩)
In the context of Perturbation Theory, state the formula for the first order correction to the energy in terms of the unperturbed wavefunctions 𝜓(n)^(0)
E(n)^(1) = ⟨ 𝜓(n)^(0) | ^H^(1) | 𝜓(n)^(0) ⟩
For systems of identical particles, describe the differences between Fermions and Bosons.
Fermions:
* The wavefunction is anti-symmetric when two particles are exchanged.
* The spin is half-integer.
Bosons:
* The wavefunction is symmetric.
* The spin is integer.
Is the electron a Fermion or a Boson?
The electron is a fermion.
What is the general formula for reduced mass?
1 / 𝜇 = ( 1 / m(1) ) + ( 1 / m(2) )
State Noether’s Theorem.
Noether’s Theorem tells us that every continuous symmetry of a Lagrangian implies the existence of a conserved quantity.
Give examples of Noether’s Theorem.
- Rotational symmetry
- Time-invariance
- Gauge-invariance
Explain the experiment by Wu.
- Wu oberserved the decay of Cobalt 60 nuclei in a magnetic field.
- This is essentially beta decay, but the electrons came out with a preferred direction.
Explain how Wu’s experiment lead to the conclusion that, in Nature, all neutrinos are left-handed and all anti-neutrinos are right-handed.
- It was inferred that the weak force does not conserve parity.
- The anti-neutrino emitted in this interaction was always right-handed in helicity.
Which gauge boson is responsible for the electromagnetic interaction?
Virtual photon, γ
Which gauge boson is responsible for the weak interaction?
W^+, W^-
Which gauge boson is responsible for the strong interaction?
pions, π^+, π^-, π^0
What are bosons?
Bosons are subatomic particles whose spin quantum number has an integer value.
Which particles are affected by the electromagnetic interaction?
Charged particles only
Which particles are affected by the weak interaction?
All types
Which particles are affected by the strong interaction?
Hadrons only
List Baryons.
- Proton: p, p^+
- Neutron: n, n^+
- Sigma: Σ^0, Σ^+, Σ^-
- Lambda: Λ^0, Λ^+, Λ^-
Does strangeness have to be conserved?
It is always conserved in strong interactions and electromangetic interactions,
but not in weak interactions.
Define the term ‘forbidden transition’.
Forbidden transitions are transitions between energy levels in a quantum-mechanical system
that are not allowed to take place because of selection rules.
Give the 5 postulates of quantum mechanics.
- The state of a system is given by the wavefunction ψ.
- Physical quantities A are represented by Hermitian Operators ^A.
- Measurement of A leads to one of the eigenvalues of ^A.
- The average outcome of an experiment is given by the expectation value.
- The wavefunction ψ evolves over time according to the time-dependent Schrodinger equation.
Classical angular momentum is defined by…
->..->..->
L = r x p
The magnitude (squared) of angular momentum for a single particle is given by…
L^2 = L(x)^2 + L(y)^2 + L(z)^2
What is the equation for an electron in an electrostatic field of nucleus?
V(r) = - (e^2)/(4πε(0)r)
Where e is the charge of the electron.
What is the fundamental equation for thermodynamics?
TdS = dU + pdV - μdN
dU = internal energy change
pdV = mechanical work done
μdN = chemical work done
(There’s Shit Under Van Now)
State the definition of a vector for parity transformations.
Any vector-like quantity that is odd under a parity transformation we call a vector.
State the definition of a pseudo-vector for parity transformations.
Any vector-like quantity that is even (i.e. does not change sign) under parity is called a pseudo-vector.
Describe how the 1D TDSE and 1D TISE equations are related.
The time-dependent equation factors in both temporal and spatial data and determines the behavior of a quantum particle over time.
The time-independent equation factors in spatial data and determines the behavior of a stationary quantum particle.
Give an overview of orbital angular momentum.
Orbital angular momentum:
* Has classical counterpart (angular momentum)
* l and m quantum numbers take on integer values
* Can be represented as operators acting on wave-function space
Give an overview of spin.
Spin:
* Intrinsic angular momentum of a particle
* No classical counterpart
* Quantum numbers l and m usually called s and m(s)
* Has to be represented as abstract matrices (‘by hand’)
What is the equation for a rigid motor?
H = L^2 / 2μR^2
Where μ is the reduced mass.
A function Ψ(x, y, z) is even if…
Ψ(-x, -y, -z) = Ψ(x, y, z)
A function Ψ(x, y, z) is odd if…
Ψ(-x, -y, -z) = -Ψ(x, y, z)
What does the HOMO level of a molecule refer to?
HOMO is an acronym for highest occupied molecular orbital.
What does the LUMO level of molecule refer to?
LUMO is an acronym for lowest unoccupied molecular orbital.
Which analogies can be made between the HOMO and LUMO levels of a molecule and the energy bands of a semiconductor?
The electronic energy levels of molecules are intermediate in character between atomic levels and the band-structure of solids/metals,
with occupied and unoccupied orbitals,
similar to the valence and conduction bands of a semiconductor.
The Lorentz force is given by:
F = qE + qv x B
State which of the terms in this equation are scalars, which are vectors and which are pseudo-vectors.
E, v and F are vectors,
B is a pseudo-vector and q and q are scalars.
The wavefunction for 4 identical fermion particles satisfies
Ψ(x₂, x₁,x₄, x₃) = ζ Ψ(x₁, x₂, x₃, x₄)
Give the value for ζ and state your reasoning.
For fermions, the wave function must be anti-symmetric under particle exchange.
Mathematically, this means that
Ψ(x₁, x₂, x₃, x₄) = -Ψ(x₂, x₁, x₄, x₃).
Therefore, in this case, ζ = -1.
The wavefunction for 4 identical boson particles satisfies
Ψ(x₂, x₁,x₄, x₃) = ζ Ψ(x₁, x₂, x₃, x₄)
Give the value for ζ and state your reasoning.
For bosons, the wave function must be symmetric under particle exchange.
Mathematically, this means that
Ψ(x₁, x₂, x₃, x₄) = Ψ(x₂, x₁, x₃, x₄).
Therefore, in this case, ζ = 1.
For a Lagrangian for a particle of mass m and charge e, what is the canonical momenta conjugate for x coordinate?
P(x) = ∂L / ∂ẋ
For a Lagrangian for a particle of mass m and charge e, what is the canonical momenta conjugate for y coordinate?
P(x) = ∂L / ∂ẏ
For a Lagrangian for a particle of mass m and charge e, what is the canonical momenta conjugate for z coordinate?
P(x) = ∂L / ∂ż
The magnitude of orbital angular momentum is given by…
L^2 = l (l + 1) ħ^2
Where l can take any non-negative integer value:
l = 0, 1, 2, 3, …
The orientation of orbital angular momentum is given by…
L(z) = mħ
Where m can attain values between -l and +l in integer steps:
m = -l, -l + 1, … , l
