3 - Quantum Dynamics

Question 1

How do you determine the time evolution of a quantum system?

Answer 1

1. Find the Hamiltonian H for your system.
2. Determine the energy eigenvalues E_n & the energy eigenvectors |E_n> by solving:

H |E_n> = E_n |E_n> n=1,2,3,…

3. Write the initial state in terms of the energy eigenbasis:

|Ψ(0)> = Σ_n c_n |E_n>

4. The state at time t is:

|Ψ(t)> = Σ_n c_n e^(-iE_nt/ℏ) |E_n>

Question 2

What is meant by spin precession?

Answer 2

When two or more spin components evolve in time with different time-dependent phase terms, a spin precession is created where the probability of measuring a particular spin component changes with time.

Question 3

What is the expression for omega_0?

Answer 3

q_e*B_0/m_e

Question 4

What is the Hamiltonian Operator?

Answer 4

The Hamiltonian operator is the energy operator. Its eigenvalues represent the possible energies of the system, and its eigenstates are states of definite energy and are therefore the stationary states under evolution. The Hamiltonian generates time evolution.

Question 5

What is Larmor frequency?

Answer 5

The Larmor or precessional frequency refers to the rate of precession of the magnetic moment of the particle around the external magnetic field. The frequency of precession is related to the strength of the magnetic field, B_0.

ω = (e * B_0) / m

Question 6

What is the uncertainty of a spin? How do you calculate it?

Answer 6

It’s just the potential difference between spin measurements.

ΔS_n = [ - ()^2]^0.5