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Engineering Physics Year 3 > Particles, Quanta & Fields > Flashcards

Particles, Quanta & Fields Flashcards

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1
Q

Give examples of an EM wave acting as a particle.

A
  • Compton scattering
  • Photoelectric effect
  • Black body radiation
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2
Q

Give an example of a particle acting as a wave.

A

Double slit experiment with electrons.

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3
Q

What is the Heisenberg Uncertainty Principle?

A

∆x ∆p ≥ ħ / 2

Where:
∆x is the uncertainty in position
∆p is the uncertainty in momentum

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4
Q

What is the 1D TDSE?

A

iħ 𝜕Ψ / 𝜕t = ( - ħ^2/ 2m ) (𝜕^2Ψ(x, t) / 𝜕x^2) + V(x)Ψ(x, t)

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5
Q

What is the 1D TISE?

A

(- ħ^2/2m) (d^2Ψ(x) / dx^2) + V(x)Ψ(x) = EΨ(x)

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6
Q

What is the variational principle?

A

The variational principle is a way to approximate the ground state.

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7
Q

Give the equation for the Hamiltonian.

A

^H = - (ħ^2 / 2m) (∂^2 / ∂x^2) + V(x)

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8
Q

What is the definition of the Hamiltonian?

A

The sum of the kinetic and potential energies of all particles associated with the system.

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9
Q

What is i^2?

A

i^2 = -1

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10
Q

If z = x + iy what is the complex conjugate of z?

A

z* = x - iy

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11
Q

What is normalisation?

A

∫ (∞, -∞) |Ψ(x)^2| dx = 1

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12
Q

What is the equation for wavenumber, k?

A

k = 2π / λ = 2πf / c

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13
Q

State what it means for a wavefunction |φ〉to be an eigenfunction of an operator ˆA.

A

|φ〉 is an eigenfunction of an operator ˆA if:

ˆA|φ = λ |φ〉

where λ is the eigenvalue corresponding to the eigenfunction |φ〉.

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14
Q

What is the dirac notation?

A

〈φ|ψ〉 =∫ (∞, −∞) φ∗ψ dx

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15
Q

How do you find the complex conjugate of something?

A

You find the complex conjugate by changing the sign of the imaginary part of the complex number.

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16
Q

What is the operator form of the Schrodinger equation?

A

^H ψ = E ψ

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17
Q

What is the momentum operator?

A

^ρ = (ħ / i) (𝜕 / 𝜕x)

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18
Q

In quantum mechanics, what are physical quantities represented by?

A

Operators

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19
Q

What is the 3D TDSE?

A

iħ 𝜕Ψ / 𝜕t = ( - ħ^2/ 2m ) ((𝜕^2 / 𝜕x^2) + (𝜕^2 / 𝜕y^2) + (𝜕^2 / 𝜕z^2))Ψ + V(x, y, z)Ψ

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20
Q

What is the 3D TISE?

A

(- ħ^2/2m) ((𝜕^2 / 𝜕x^2) + (𝜕^2 / 𝜕y^2) + (𝜕^2 / 𝜕z^2))ψ + V(x, y, z)ψ = Eψ

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21
Q

What is the 3D Hamiltonian?

A

^H = - (ħ^2 / 2m) ((𝜕^2 / 𝜕x^2) + (𝜕^2 / 𝜕y^2) + (𝜕^2 / 𝜕z^2))+ V(x, y, z)

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22
Q

What is the expectation value definition?

A

The expectation value is the probabilistic expected value of the result (measurement) of an experiment.

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23
Q

What is the expectation value?

A

⟨A⟩ = ⟨ψ|A|ψ⟩

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24
Q

An operator is called Hermitian if…

A

⟨𝜙 ∨ ^A 𝜓⟩ = ⟨^A 𝜙 ∨ 𝜓⟩

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25
Q

Of which operators are the spherical harmonics Y(lm) eigenfunctions? What are the corresponding eigenvalues?

A

^(hat)L^2 whose eigenvalue is l (l + 1) ħ^2

^(hat)L(z) whose eigenvalue is mħ

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26
Q

In the context of Perturbation Theory, state the formula for the first order correction to the energy in terms of the perturbing Hamiltonian ^H

A

E(n)^(1) = (⟨ 𝜓(n)^(0) | ^H^(1) | 𝜓(n)^(0) ⟩) / (⟨ 𝜓(n)^(0) | 𝜓(n)^(0) ⟩)

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27
Q

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)

A

E(n)^(1) = ⟨ 𝜓(n)^(0) | ^H^(1) | 𝜓(n)^(0) ⟩

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28
Q

For systems of identical particles, describe the differences between Fermions and Bosons.

A

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.

29
Q

Is the electron a Fermion or a Boson?

A

The electron is a fermion.

30
Q

What is the general formula for reduced mass?

A

1 / 𝜇 = ( 1 / m(1) ) + ( 1 / m(2) )

31
Q

State Noether’s Theorem.

A

Noether’s Theorem tells us that every continuous symmetry of a Lagrangian implies the existence of a conserved quantity.

32
Q

Give examples of Noether’s Theorem.

A
  • Rotational symmetry
  • Time-invariance
  • Gauge-invariance
33
Q

Explain the experiment by Wu.

A
  • 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.
34
Q

Explain how Wu’s experiment lead to the conclusion that, in Nature, all neutrinos are left-handed and all anti-neutrinos are right-handed.

A
  • It was inferred that the weak force does not conserve parity.
  • The anti-neutrino emitted in this interaction was always right-handed in helicity.
35
Q

Which gauge boson is responsible for the electromagnetic interaction?

A

Virtual photon, γ

36
Q

Which gauge boson is responsible for the weak interaction?

A

W^+, W^-

37
Q

Which gauge boson is responsible for the strong interaction?

A

pions, π^+, π^-, π^0

38
Q

What are bosons?

A

Bosons are subatomic particles whose spin quantum number has an integer value.

39
Q

Which particles are affected by the electromagnetic interaction?

A

Charged particles only

40
Q

Which particles are affected by the weak interaction?

A

All types

41
Q

Which particles are affected by the strong interaction?

A

Hadrons only

42
Q

List Baryons.

A
  • Proton: p, p^+
  • Neutron: n, n^+
  • Sigma: Σ^0, Σ^+, Σ^-
  • Lambda: Λ^0, Λ^+, Λ^-
43
Q

Does strangeness have to be conserved?

A

It is always conserved in strong interactions and electromangetic interactions,

but not in weak interactions.

44
Q

Define the term ‘forbidden transition’.

A

Forbidden transitions are transitions between energy levels in a quantum-mechanical system

that are not allowed to take place because of selection rules.

45
Q

Give the 5 postulates of quantum mechanics.

A
  1. The state of a system is given by the wavefunction ψ.
  2. Physical quantities A are represented by Hermitian Operators ^A.
  3. Measurement of A leads to one of the eigenvalues of ^A.
  4. The average outcome of an experiment is given by the expectation value.
  5. The wavefunction ψ evolves over time according to the time-dependent Schrodinger equation.
46
Q

Classical angular momentum is defined by…

A

->..->..->
L = r x p

47
Q

The magnitude (squared) of angular momentum for a single particle is given by…

A

L^2 = L(x)^2 + L(y)^2 + L(z)^2

48
Q

What is the equation for an electron in an electrostatic field of nucleus?

A

V(r) = - (e^2)/(4πε(0)r)

Where e is the charge of the electron.

49
Q

What is the fundamental equation for thermodynamics?

A

TdS = dU + pdV - μdN

dU = internal energy change
pdV = mechanical work done
μdN = chemical work done

(There’s Shit Under Van Now)

50
Q

State the definition of a vector for parity transformations.

A

Any vector-like quantity that is odd under a parity transformation we call a vector.

51
Q

State the definition of a pseudo-vector for parity transformations.

A

Any vector-like quantity that is even (i.e. does not change sign) under parity is called a pseudo-vector.

52
Q

Describe how the 1D TDSE and 1D TISE equations are related.

A

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.

53
Q

Give an overview of orbital angular momentum.

A

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

54
Q

Give an overview of spin.

A

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’)

55
Q

What is the equation for a rigid motor?

A

H = L^2 / 2μR^2

Where μ is the reduced mass.

56
Q

A function Ψ(x, y, z) is even if…

A

Ψ(-x, -y, -z) = Ψ(x, y, z)

57
Q

A function Ψ(x, y, z) is odd if…

A

Ψ(-x, -y, -z) = -Ψ(x, y, z)

58
Q

What does the HOMO level of a molecule refer to?

A

HOMO is an acronym for highest occupied molecular orbital.

59
Q

What does the LUMO level of molecule refer to?

A

LUMO is an acronym for lowest unoccupied molecular orbital.

60
Q

Which analogies can be made between the HOMO and LUMO levels of a molecule and the energy bands of a semiconductor?

A

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.

61
Q

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.

A

E, v and F are vectors,
B is a pseudo-vector and q and q are scalars.

62
Q

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.

A

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.

63
Q

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.

A

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.

64
Q

For a Lagrangian for a particle of mass m and charge e, what is the canonical momenta conjugate for x coordinate?

A

P(x) = ∂L / ∂ẋ

65
Q

For a Lagrangian for a particle of mass m and charge e, what is the canonical momenta conjugate for y coordinate?

A

P(x) = ∂L / ∂ẏ

66
Q

For a Lagrangian for a particle of mass m and charge e, what is the canonical momenta conjugate for z coordinate?

A

P(x) = ∂L / ∂ż

67
Q

The magnitude of orbital angular momentum is given by…

A

L^2 = l (l + 1) ħ^2

Where l can take any non-negative integer value:
l = 0, 1, 2, 3, …

68
Q

The orientation of orbital angular momentum is given by…

A

L(z) = mħ

Where m can attain values between -l and +l in integer steps:
m = -l, -l + 1, … , l

Engineering Physics Year 3 (5 decks)

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