17. Hermitian Operators and Commutators

Question 1

Give the equation for the momentum of the x component of a plane wave

Answer 1

p_x = h_bar * k

Question 2

What can be said about Δk for a plane wave?

Answer 2

It is > 0 due to its Gaussian spread of possible momentum.

- Could be described as a delta function limit

Question 3

Define box normalisation for a plane wave

Answer 3

Where we pretend x + L = x. (large x)

Question 4

See page 9 of document for the mathematical representation of a general quantum state and a Hermitian operator

Answer 4

Do it

Question 5

Define Hermiticity

Answer 5

When an operator is its own complex conjugate

Question 6

What is a consequence of hermiticity?

Answer 6

All eigenvalues are real

Question 7

See page 9 of document for Hermitictity

Answer 7

Do it

Question 8

What is a consequence of hermiticity?

Answer 8

All eigenvalues are real

Question 9

Define what is meant when two things commute

Answer 9

When the order of multiplication does not matter e.g.

f(x) g(x) = g(x) f(x)

Question 10

See page 10 of document for mathematical representation of operator commutation

Answer 10

Do it

Question 11

Whatis the consequence if two operators do commute?

Answer 11

There is no uncertainty principle connecting them

- Can know them both precisely and simultaneously

Question 12

Whatis the consequence if two operators do not commute?

Answer 12

There is uncertainty principle connecting them

- Cannot know them both precisely and simultaneously

Question 13

What can be said about two operators, A and B and their eigenfunctions if they commute?

Answer 13

They should share a set of eigenfunctions

Question 14

What can be said about two operators if they share a set of eigenfunctions?

Answer 14

They commute