2. Relativistic Wave Equations

Question 1

How to we get the free field SE?

Answer 1

Set the Hamiltonian just equal to the kinetic energy, and quantise the momentum

Question 2

In the general SE, how is the time dependence of the system given?

Answer 2

By the Hamiltonian

Question 3

Is the far field SE relativistically invariant?

Answer 3

No, it still has second order derivatives of space and first order of time

Question 4

Why did Schrodinger reject the KE?

Answer 4

Because although it was relativistically invariant, it didn’t describe the energy levels in Hydrogen atoms

Question 5

What type of particles can the KG equaiton be used to describe?

Answer 5

Spinless particles

Question 6

How do we obtain the conserved current when using the KG equation?

Answer 6

Start with the KG
- Multiply left of eqn by a plane wave and the right by its conjugate
- Do the same in the opposite order for another equation
- Work through and use the product rule for the operators

Question 7

What problem arises when looking at the solution of the 4 current of the KG equation

Answer 7

The 0 component which represents the probability density is proportional to the energy
- We already know that there can be negative energy solutions and negative probability makes no sense

Question 8

Summarise the 5 problems with the KG equation

Answer 8

• Simplest solutions are scalars which don’t account for spin
• Fails to describe the hydrogen atom
• There are positive and negative energy solutions
• The prob. density is not positive-definite
• Equation is second order in time like derivatives, so we need BCs everywhere in space

Question 9

How is electromagnetism introduced into the KG equation?

Answer 9

By minimal substitution using the 4 potential A^mu

Question 10

When looking at the solutions for the KG equation, what do we know about our choice of solutions for alpha and beta?

Answer 10

They must be 4x4 matricies as they anti commute
- Obvious candidate are the Pauli matricies

Question 11

What does U represent?

Answer 11

The unitary matrix

Question 12

What is a unitary matrix?

Answer 12

One where its inverse is equal to its conjugate transpose

Question 13

What are the gamma matricies?

Answer 13

γ^mu = (γ^0, γ^i)
- It is not a 4 vector as the components are matricies

Question 14

State the expansion for when you calculate the commutation and anticommutation for two operators, A and B:

Answer 14

[A, B] = AB - BA
{A, B} = AB + BA

Question 15

Are the Hamiltonian eigenvalues real or imaginary?

Answer 15

Real, so the conjugate transpose (H dagger) is equal to H

Question 16

What is the trace of a matrix, and what is it equal to for the alpha, beta and gamma matricies equal to?

Answer 16

The trace of a matrix Tr(M) is summing the components on the diagonal
- For the matricies, they all are traceless

Question 17

What is the cyclic property of taking the trace of a matrix, and what is Tr(A+B) equal to?

Answer 17

Tr(CAB) = Tr(ABC)

Question 18

What is a consequence of the alpha, beta and gamma matricies all being traceless?

Answer 18

The Hamiltonian is traceless
- Implies the energy eigenvalues sum to 0
- Back to positive and negative energy solutions

Question 19

How is a Clifford Algebra Basis generated?

Answer 19

By all independent products of generators of the algebra

Question 20

How is the Clifford algebra defined for 2x2 matricies?

Answer 20

As linear combinations of the Pauli matricies

Question 21

State the properties of the Weyl spinors, and which particles do they describe?

Answer 21

They have fixed handedness and violate parity
- They describe massless fermions
- Mass leads to mixing together of the massless solutions
- A Dirac Spinor can be formed from a pair of Weyl spinors

Question 22

What is the defining characteristic of the gamma matrix?

Answer 22

Their anti commutation
{γ^mu, γ^ nu} = 2g^mu nu