Relativistic Kinematics, Classical Electromagnetism and Quantum Mechanics Flashcards

1
Q

Invariant Mass Calculations

Overview

A

-looking for evidence of a new particle X
-suspect that X decays to A+B
-X is short-lived so can’t observe it directly
-in experiment, collide e- and e+ and regularly observe:
e- + e+ -> A + B (+ other stuff)
-want to know if the A and B come from X
-plot the relative frequency of invariant mass of A and B and see if there is a peak at mx, the mass of X

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Invariant Mass

Definition

A

W² = (ΣEi)² - (Σpi_)²

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Wavefunction of X Particle

A

ψ(t) = ψ(0) e^(-iEt) e^(-Γt/2)
-for a particle at rest, E=mx:
ψ(t) = ψ(0) e^(-iEmx) e^(-Γt/2)
-the second exponential is the real part required to ensure an overall decrease in probability over time since the particle is likely to decay
-Γ is the decay constant, factor of 1/2 since the probability equals |ψ|² so factor of two cancels

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Breit-Wigner Formula

A

ϱ(E) = R / [(E-mx)² + Γ²/4]

-Γ is the full width half maximum

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

If A and B haven’t come from X decay, what does W represent?

A

-the effective mass at the centre of mass of the composite system of A and B

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

Derive the Maxwell Equation

A
-combine the field strength tensor:
Fμν = ∂μAν - ∂νAμ
-and 
∂μFμν = Jν
=>
∂μ∂μAν - ∂μ∂νAμ = Jν
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Derive the Charge Continuity Equation

A
-start with the Maxwell Equation
∂μ∂μAν - ∂μ∂νAμ = Jν
-differentiate with respect to xν:
∂ν∂μ∂μAν - ∂ν∂μ∂νAμ = ∂νJν
-differentials commute so LHS is 0:
∂νJν = 0
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

Derive Charge Conservation

A

-start with the continuity equation:
∂νJν = 0
-integrate over some volume, V
-split the differential into the 0th element and 1,2,3 elements
-apply divergence theorem to RHS and definition of charge on LHS:
∂Q/∂t = - ∫ Ji dsi
-i.e. charge in a region only changes if current density flows across the boundary, charge is conserved
-in particular if V is the universe then total charge Q is constant

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Gauge Invariance

Definition

A

-the Maxwell Equation is invariant under transformations of the form:
Aμ -> Aμ’ = Aμ + ∂μχ
-for any scalar function χ
-check by subbing into the Maxwell equation, terms cancel and the original equation is recovered
-can also show that Fμν=Fμν’

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Gauge Invariance

Fixing the Gauge

A
  • in practice, we must choose some condition that A satisfies in order to ‘fix the gauge’
  • i.e. a condition that uniquely defines A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Coulomb Gauge

A

∂iAi = 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

Lorenz Condition

A

-not technically a gauge as it only restricts A and doesn’t uniquely define it
∂μAμ = 0
-equivalent to:
∂ . A = 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

Maxwell Equation Under the Lorenz Transformation

A

∂²Aμ = Jμ

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

Quantum Mechanics

Wave Mechanics Formulation

A

-the wave mechanics formulation of quantum mechanics postulates a wave function to describe the system
-each observable quantity from classical mechanics is promoted to an operator
-making a measurement of an observable results in an eigenvalue and immediately after measurement the system takes on the corresponding eigenstates
-between measurements the wavefunction evolves
-in non-relativistic quantum mechanics this evolution is described by the time dependent Schrodinger equation:
i ∂/∂t Ψ = -1/2m ∂i∂iΨ + VΨ = H^Ψ

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

Where does the time dependent Schrodinger equation come from?

A

-in natural units, quantum theory equates energy of a particle with frequency and momentum with wave vector:
E=ω
p_=k_
-assuming the time-evolution equation to be linear , any wavefunction can be constructed out of a complete set of plane-wave solutions as they form a basis
-so can deduce an appropriate equation by looking at plane-wave solutions:
Ψ = Ψoe^[ip_.x_] = Ψoe^[i(Et-pxx-pyy-pzz)]
-differentiate with respect to time to find the energy operator
-take the gradient to find the momentum operator
-sub into energy = KE + PE equation for Schrodinger equation

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

Probability Amplitude

A

-the interpretation of the wavefunction is as a probability amplitude
-the probability of finding a particle described by a wavefunction Ψ in a region V is given by:
P(V) = ∫ |Ψ(x)|² d³x

17
Q

Probability Density Current

Definition

A

probability density only changes in a region if:
ji = -1\2mi (Ψ∂iΨ - Ψ∂iΨ)
-the probability density current flows through the surface of that region

18
Q

Probability Density Current

Derivation

A

-multiply Schrodinger equation by Ψ* (1)
-take conjugate of Schrodinger equation, then multiply by Ψ (2)
-subtract: (1) - (2)
=>
∂/∂t(|Ψ|²) = -1\2mi ∂i (Ψ∂iΨ - Ψ∂iΨ)
=>
ji = -1\2mi (Ψ∂iΨ - Ψ∂iΨ)

19
Q

Quantum Mechanics of Charged Particles

A
  • charged particles in the presence of an electromagnetic field requires a particular form of the Schrodinger equation
  • in this case want a Hamiltonian that gives the Lorentz Force Law from the Hamilton equations
20
Q

Hamilton Equations

A

x’ = ∂H/∂pi
p’i = - ∂H/∂xi
-these equation are completely general

21
Q

Lorentz Force Law

A

Fi = Q (Ei + εijk x’j Bk)

22
Q

Hamiltonian From the Lorentz Force Law

A

-write the Lorentz Force Law in terms of electromagnetic potentials
-swap εijk for deltas
-sub in dA/dt
=>
Fi = Q(-∂iV - dA/dt + x’j∂iAj)

23
Q

Electromagnetic Form of the Schrodinger Equation

A

i ∂Ψ/∂t = δij(pi-QAi)(pj-QAj)/2m Ψ + VQΨ

-recognise that the derivatives act on A as well as Ψ

24
Q

Classical Angular Momentum

A

Li = εijk rj pk

25
Quantum Angular Momentum Operator Components
Li^ = -i εijk rj ∂k
26
Angular Momentum Commutation Relations
``` [L1^,L2^] = iL3^ [L2^,L3^] = iL1^ [L3^,L1^] = iL2^ ```
27
Total Angular Momentum Operator
L²_^ = L1²^ + L2²^ + L3²^ | -it can be shown that this commutes with Li^
28
Angular Momentum Eigenvalues
- boundary conditions on on the eigenstates of L_^ and Li^ impose certain allowed values / quantum numbers, specifically: - L²_^ has eigenvalues l(l+1) for l∈ℕ - Li^ has eigenvalues m∈Z such that -l≤m≤l
29
Angular Momentum Operator | Simultaneous Eigenstates
-can be in simultaneous eigenstate of L_^ and Li^ but NOT e.g. L1^ and L2^
30
Intrinsic Angular Momentum
- intrinsic angular momentum is spin - the boundary condition argument does not apply in this case since spin is an intrinsic property - introduce spin operators, Si^ and S^_² that obey the same commutation relationships as the angular momentum operators
31
Spin Operator Eigenvalues
S^_²ψ = s²ψ e.g. Si^ψ = msψ
32
Ladder Operators for Spin
-consider a simultaneous eigenstate of S3^ and S^_², ψ, with eigenvalues ms and s² -construct ladder operator: S±^ = S1^ ± iS2^
33
Spin Ladder Operators | Commutation Relations
1) [S3^,S±^] = ± S±^ 2) [S+,S-] = 2 S3^ 3) [S±^,S^_²] = 0 - (1) => these are ladder operators for S3^: S+ ψ is an S3^ eigenstate with eigenvalue ms+1, similarly acting with S-^ gives eigenvalue ms-1 - (3) => fact that S±^ commute with S^_² tells us that when we change S3^ value we still have S^_²eigenstate with the same value, i.e. changing third component of angular momentum you don't change overall angular momentum
34
Total Angular Momentum and S+ & S-
``` -for given total spin, S±^ gives us all possibilities of S3^ S^_² = S1^² + S2^²+ S3^² = 1/2 (S+S- + S-S+) + S3^² = S3^ + S-^S+^ + S3^² -OR = -S3^ + S-^S+^ + S3^² ```
35
Angular Momentum | ψmax
-let ψmax be the state with the maximum possible ms for given S -know that: S3^S+^ ψmax= (mmax + 1)S+ ψmax -this must hold but also have a maximum therefore S+^ψmax must =0 -S+^ annihilates ψmax -similarly S-^ annihilates ψmin
36
Relation Between Total Spin and Maximum Eigenvalue for One Spin Component
``` S^_² ψmax = (S3^ + S-^S+^ + S3^²) ψmax = (mmax + mmax²) ψmax = mmax ( 1+mmax) ψmax => s = √[mmax(mmax+1)] -AND s = √[mmin(mmin-1)] ```
37
Allowed Values of Spin
``` -putting the two expressions for s in terms of mmax and mmin together => mmin = ±mmax -and m's differ by an integer amount, so: mmax = {0,1/2,1,3/2,3,...} -spin statistics theorem => integer spin = bosons half integer spin = fermions ```
38
Schrodinger Equation for an Electron
-an electron is spin 1/2 so need two degrees of freedom, spin up or spin down (1/2,-1.2) -we promote ψ to a spinor-valued wave function with two components for the two spin states -with two component spinors, can write 2x2 representation of spin operators: si = 1/2 σi -where σi are the Pauli matrices -these ideas can be incorporated straight into the Schrodinger Equation to form the Pauli Equation, a non-relativistic equation for a spin 1/2 charged particle