21. Time Dependence of Expectation Values

Question 1

What is the rate of change of the expectation value of an observable associated with a time independent operator proportional to?

Answer 1

The commutator with the Hamiltonian

Question 2

See p13 of document for a mathematical representation of the rate of change of the observable and the Hamiltonian

Answer 2

Do it

Question 3

What can be said if the commutator [ H, Q] = 0?

Answer 3

The operator commutes with the Hamiltonian

• The rate of change of the expectation value of the observable is 0
• > Dynamical variable is conserved

Question 4

See page 13 of document for the commutator between the Hamiltonian and the 1d momentum operator

Answer 4

Do it

Question 5

What classical law is recovered when commuting the Hamiltonian and the 1d momentum operator?

Answer 5

Newton’s second law

Question 6

What classical law is recovered when commuting the KE and the 1d momentum operator?

Answer 6

p = mv

Question 7

Does the potential operator commute with the position vector? / are they compatible

Answer 7

Yes provided V(x) ONLY

- We can know them both precisely and simultaneously

Question 8

What is the variance of the Hamiltonian operator?

Answer 8

0

Question 9

See page 14 of document for a mathematical proof that the variance of the Hamiltonian is 0

Answer 9

Do it

Question 10

From the variance proof on page 14 for the Hamiltonian, why is the variance of the energy equal to 0?

Answer 10

• ψ is a stationary state
• V(r) ONLY
  Energy is constant
  Any measurements of E give exactly E as expected