Class 11

• Loose ends: Convolution Theorem (see Steck, section 1.A.2); units of frequency, angular frequency, and decay rate.
  
  
• Review of quantum mechanics:
  
  
  • the quantum state of a system, represented as a complex vector in an infinite-dimensional linear vector space (the ket vector).
    
    
  • the scalar product (complex inner product) of state vectors: dual vector spaces and the bra vector.
    
    
  • the representation of dynamical observables as operators on the quantum state; examples of position, momentum, angular momentum, kinetic energy, and total energy operators, expressed in the real-space basis (i.e., as differential operators).
    
    
  • eigenfunctions of the angular momentum operators, L² and L_z: the spherical harmonics.
    
    
  • plane-wave expansion of the wavefunction as an example of superposition.
    
    
  • eigenstates and eigenvalues of operators; expansion of the quantum state as a superposition of eigenstates.
    
    
  • expectation values of dynamical observables for a specified quantum state.
    
    
  • Hamiltonian operator and its connection to the time evolution of the state of a system: the time-dependent wave equation.
    
    
  • Formal solution of the wave equation for constant-time Hamiltonian; interpretation of the exponential of an operator.
    

Class 12

• Representation of the ket vector in the coordinate basis, and its relation to the wavefunction in coordinate space.
  
  
• Extension of the state space to include the spin observable for a spin-½ particle: tensor product of ket vectors, and the expansion of the state vector in the higher dimensional state space.
  
  
• Correlation between spatial coordinates and spin: entanglement of dynamical observables; factorization of the state vector into a tensor product, in the absence of entanglement.
  
  
• Three forms for the representation of operators:
  
  • Coordinate basis: differential operators and functions of the position vector.
  • Arbitrary basis: matrix representation; matrix elements of an operator; diagonal form of matrix in the operator's eigenvector basis.
  • Arbitrary basis: superposition of products of bra and ket vectors; forms of the operator in the eigenvector basis, and under an arbitrary basis.
  
  
  
• Projection operators.
  

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

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

Class 13

• Review of HW6, absorption by ⁸⁷Rb vapor: comparison of calculated absorption for a 3 cm path with experimental values of transmittance for the two broad resonances; energy levels for the ⁸⁷Rb ground state.
  
  
• Transformation of the state vector from one orthonormal basis set to another: the composition rule for transforming the superposition coefficients.
  
  
• Degenerate eigenvectors and the complete specification of the state of a quantum system: simultaneously measurable variables and the requirement for commuting operators; enlargement of the state space through tensor products of the eigenvectors of a complete set of operators.
  
  
• Simple model for the state space of a one-electron atom: set of 7 operators, H, L², L_z, S², S_z, I², I_z, their corresponding eigenvalues, allowed quantum numbers (n, l, m_l, s, m_s, I, m_I), the tensor product basis vectors, and the expansion of the state in this basis set.
  

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

• Evolution of the state vector for a Hamiltonian which is constant in time: solutions and equations of motion for the time-dependent superposition coefficients; probability of finding the system in a given eigenstate is constant in time: | c_i(t) |² = | c_i(0) |²
  
  
• Magnetic moment of a particle with angular momentum; the spin vector operator; the S_z operator in matrix form and as a superposition of projection operators; eigenvalues and eigenvectors of S_z; the S_x and S_y operators; representation of the state vector as a superposition of eigen ket vectors, and as a column vector; representation of the bra vector as a row vector.
  
  
• Example (cf. section 2.9.2 of Demtröder): a spin-½ particle in a constant, uniform magnetic field; Hamiltonian operator, Larmor frequency, and eigenvalues of H; the time-dependent state vector; expectation value of the spin vector operator; calculation of ⟨ S_z ⟩ and its constancy in time.
  

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

HW9: The expectation value of ⟨ S_z ⟩ 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 ⟨ S_x ⟩ and ⟨ S_y ⟩.

b) Describe the motion of the vector formed by the components ( ⟨ S_x ⟩, ⟨ S_y ⟩, ⟨ S_z ⟩ ), i.e. the expectation value of the spin vector.

Class 16

• Corrections to previous notes on Hamiltonian for electron in uniform B-field, and definition of Larmor frequency (ω_L); g-factor for electron.
  
  
• Time-dependent Hamiltonians: Hamiltonian as sum of time-independent and time-dependent parts, H = H₀ + V(t); representation of state vector in the H₀ eigenvector basis; equations of motion for the superposition coefficients.
  
  
• Example: Spin-½ particle in a rotating magnetic field (cf. exercise 16.23 in Merzbacher, and Rabi's 1937 paper); the Hamiltonian operator for this system; application of the general equations of motion for this problem.
  

Class 17

• Review of solutions to HW8 and 9; direction of precession for an electron in a magnetic field (from HW 9b).
  
  
• Magnetic dipole moment of the proton; energy level splitting in an external magnetic field; Larmor frequency of the proton in a 1 Tesla field; magnetic resonance imaging: relation between the typical field strength and the Boltzmann factor.
  
  
• The time-dependent equations of motion for the superposition coefficients in a constant potential: compare the expansion of the state in the eigenvector basis of H₀ vs the eigenvector basis of constant-time H; transitions induced by a constant potential (example of spin-½ particle in a constant B-field along z, with small constant component along x).
  
  
• Solution of the equations of motion for the spin-½ particle in a rotating magnetic field (from last class); specific solution for the case c₁(t = 0) = 1; expression for the probability of finding the particle in the “spin-up” state, | c₁(t) |²; solution for | c₁(t) |² and | c₂(t) |² on resonance (ω = ω_L); behavior in the limit of large detuning (ω ≫ ω_L).
  

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 S_z 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 S_z) vs a measurement of the spin component along the direction of the B field.

Class 18

• Transition probability for a spin-½ particle in a rotating magnetic field: P(½, −½) for co-rotating and counter-rotating cases, on resonance; ineffectiveness of the counter-rotating field for inducing transitions; dependence of transition probability on angle of field with respect to z-axis.
  
  
• Atomic parameters: the Bohr radius (a₀); expectation value, ⟨ r ⟩, for the ground state of the H atom; connection between the expectation value in the state space description and in the wavefunction description; order of magnitude estimate for the Coulomb field strength seen by the atomic electron.
  
  
• Order of magnitude estimates for the field strength of a cw laser, unfocused, and focused to a diffraction-limited spot.
  

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

• Solutions for | c₁(t) |² and | c₂(t) |², for the case of a spin-½ particle in a purely transverse (θ = π / 2) B-field.
  
  
• Hamiltonian for a charged particle in a time-varying electromagnetic field:
  • the role of the Hamiltonian function in classical dynamics: Hamilton's equations of motion.
    
    
  • equation of motion for a charged particle in an EM field.
    
    
  • relations between the E and B fields to the scalar and vector potentials, φ(r, t) and A(r, t).
    
    
  • classical Hamiltonian function for charge q, with mass m, in time-varying vector and scalar potentials (c.f. Goldstein, ch. 1 and 8); interpretation of the momentum as the dynamical momentum of the particle plus the momentum within the field produced by the particle.
    
    
  • elevation of the classical Hamiltonian function to the Hamiltonian operator, H, in the semi-classical description.
    
    
  • role of the vector and scalar potentials in quantum mechanics: the Aharanov-Bohm effect.
    
    
• Separation of the H into the sum of a time-independent atomic Hamiltonian (H₀) and a time-dependent potential energy operator, V(t); V(t) in the Coulomb gauge; justify dropping the A² term in V(t) by estimating the ratio of the A² to A·p terms of V(t) at moderate laser field strengths.
  
  
• V(t) = (e/m) A·p replaced by −(d·E), where d is the dipole moment vector operator: d = −er.
  
  
• Electric dipole approximation: justification for ignoring spatial variation of the E field by estimating variation of k·r across the size of an atom.
  

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, ω_L).

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

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

Class 20

• Matrix elements of the interaction potential for a monochromatic wave with atomic electron: role of the matrix elements of r (see exercise).
  
  
• Expectation value of r for a two-level atom; parity of eigenstates in a spherically symmetric potential, and its consequence for the matrix elements of r; expectation value of dipole moment for atom in an eigenstate vs a superposition state.
  
  
• Matrix representation of H₀ and V(t) for a two-level atom in a monochromatic, optical field.
  
  
• Application of the general QM equations of motion to the two-level atom driven by a monochromatic field (eqns. 2.68 in Demtröder); comparison with the equations of motion for a spin-½ particle in a rotating magnetic field.
  

Exercise:

a) Prove that the momentum operator p is given by

---

Last revised on Saturday, 11-Nov-2017 10:30:47 EST
©1998--2016 Creative Consulting for Research and Education