Laser Spectroscopy (Classes 11 through 20)

Copyright © 2009 Krishna Myneni

Classes:  11  12  13  14  15  16  17  18  19  20

Homework:  8  9  10  11

Class 11

Class 12

Exercise: Show that an operator, B, may be represented in a complete ortho-normal basis as

B = ∑i,j ( Bij | i ⟩ ⟨ j | )

where Bij are the matrix elements of B.

Exercise: Show that the sum over all projection operators, in an orthonormal basis suitable for representing the quantum state, is the identity operator:

i ( | i ⟩ ⟨ i | ) = 1

Class 13

HW8: In the simplified state space for the one-electron atom, described above, the principal quantum number, n, has a range of one to infinity. Consider a truncated state space (subspace) where n has a range of one to a maximum value N.

a) If the nuclear spin quantum number I = 0, what is the number of dimensions in the subspace, expressed in terms of N?

b) For N = 2, 10, and 100, compute the number of dimensions in the subspace. The number of dimensions is the number of complex values needed to represent the state vector within the subspace. Note that N = 2 does not represent an ideal two-level system.

c) How is the result of a) modified for an atom with nuclear spin, I ?

d) For the 133Cs D1 and D2 transitions, the eigenvectors involved are those with quantum numbers n = 6, l = 0, and l = 1. The nuclear spin is I = 7/2. Find the number of eigenvectors needed to describe the state of the atom for the D1 and D2 transitions.

Hint: For part a) you will need formulae for the sums of the integers (n) and sums of squares of integers (n2).

Class 14

Midterm Exam

Class 15

Exercise: Given the representation of Sx and Sy below, find the corresponding matrix representations of these operators in the Sz eigenvector basis, i.e. in the basis set { |+⟩, |−⟩ }:

Sx = (ℏ/2) ( |+⟩ ⟨−|  +  |−⟩ ⟨+| )

Sy = (i ℏ/2) ( |−⟩ ⟨+|   |+⟩ ⟨−| )

HW9: The expectation value of ⟨ Sz ⟩ was computed in class, for the case of a spin-½ particle in a constant uniform magnetic field along the z-axis.

a) Find the expectation values ⟨ Sx ⟩ and ⟨ Sy ⟩.

b) Describe the motion of the vector formed by the components ( ⟨ Sx ⟩, ⟨ Sy ⟩, ⟨ Sz ⟩ ), i.e. the expectation value of the spin vector.

Class 16

Class 17

Exercise: For a B-field of 1.5 Tesla, find the relative population of the spin-up and spin-down states for protons at T = 300K.

Exercise: For the spin-½ particle in a rotating magnetic field, starting from the coupled equations of motion, fill in the steps of the derivation for the solution of the time-dependent superposition coefficient, c1(t).

HW10: For the case of the spin-½ particle in a constant B-field, with a strong z-axis component, and a weaker x-axis compent, find the solutions to the superposition coefficients in the Sz eigenvector basis:

a) Write down the Hamiltonian for this system as H = H0 + V, where H0 is the Hamiltonian for the interaction of the particle with only the z-component of the field. What is the matrix representation of V?

b) Write down the equations of motion in the H0 basis, using the general result for time-dependent problems.

c) Solve the equations of motion to find c1(t) and c2(t).

d) Describe the difference in measurement of the spin component along the z-axis (measurement of Sz) vs a measurement of the spin component along the direction of the B field.

Class 18

Exercise: Transform the expression for the transition probability, P(½, −½), to a function of θ, the angle of the field w.r.t. the z-axis.

Exercise: Show that the expectation value ⟨ r ⟩ for the ground state of the H atom is, ⟨ r ⟩ = (3/2) a0, where a0 is the Bohr radius. The normalized wavefunction for the | 1s ⟩ state was given in class.

Class 19

HW11: For the spin-½ particle in a co-rotating B-field, find ⟨ S ⟩, and

a) Describe the motion of ⟨ S ⟩ near resonance (ω approximately equal to the Larmor frequency, ωL).

b) Describe the motion of ⟨ S ⟩ at large detunings.

Plot ⟨ Sz ⟩ vs time to help you visualize the motion for each case.

Class 20


a) Prove that the momentum operator p is given by

p = m [r, H0] / (i ℏ)


H0 = p2 / (2m) + V0

and given that r and V0 commute, i.e. [r, V0] = 0.

b) Using a), show that the matrix elements of A·p are proportional to the matrix elements of r:

⟨ i | A·p | j ⟩ = m (λj - λi) A·⟨ i | r | j ⟩ / (i ℏ)

where | i ⟩ and | j ⟩ are eigenvectors of H0, and λi, λj are the corresponding eigenvalues.

Hint: In the semiclassical approximation, A(r,t) is simply a vector field, and does not act as an operator.