Spring 2024 Homework Assignments
Consider a pair of particles: first a spin 3/2 particle, second a spin 1/2 particle. (a) Starting with the knowledge that the outer product state |3/2 3/2⟩ |1/2 1/2⟩ is the same as the total angular momentum state |2 2⟩, use the lowering operator on these quantities to obtain the outer product expansions for all |2 m⟩ total angular momentum states. For the rest of the parts of this question, use the table for Clebsch-Gordan coefficients: (b) If is known that the total angular momentum state of the system is |1 0⟩, what are the possible values and their probabilities for a measurement of the z-component of the angular momentum of the spin 3/2 particle? (c) What are the corresponding quantities for the spin 1/2 particle? (d) For the particles above, if it is known that the particles are in the outer product state |3/2 1/2⟩ |1/2 -1/2⟩ what are the possible values and their probabilities for a measurement of the total angular momentum of the particles?
Find ⟨r⟩, the average of the radial distance r, for the Hydrogen states with corresponding radial functions (a) R30 (a) R31 (a) R32.
Two non-interacting electrons are placed a one-dimensional harmonic oscillator potential. One electron is in the ground state, while the second one is in the n'th energy level (with n=0 or 1 or 2, etc.) Find ⟨ (x1-x2)2 ⟩, the average of the square of the distance between the particles. Obtain results for values of n, with the spins of the particles in (a) the singlet state (b) the triplet state.
Consider a quantum system with three energy levels ε0=0, ε1=ε and ε2=2ε where ε is a constant value of energy. Suppose that three non-interacting particles are placed in this system. Discuss separately, when the three particles are identical Bosons or if they are identical Fermions, (a) the total energy (Etotal) states and their degeneracies, (b) the average value of the energy (Eaverage) as a function of temperature. Remebering that the probability of occupation of these states will be proportional to exp( -Etotal/kBT), (c) provide computer plots of Eaverage/ε versus kBT/ε and discuss the T→0 and T→∞ limits.
Two spin 1/2 Fermions are placed in a one-dimensional box of width a and infinite potential outside. A repulsive interaction (which is to be treated as a perturbation) exists between the particles: H(1) = αδ(x1-x2) where x1 and x2 are the coordinates of the two particles. One of the particles is in the n'th while the other is in the m'th energy state with m≠n. Find the first order change in the energy of the state when (a) the particles are in the singlet spin state, (b) the particles are in a triplet spin state. Find the second order change in the energy of the state when (c) the particles are in the singlet spin state, (d) the particles are in a triplet spin state.
The Hamiltonian of a three-state system has the matrix elements ( 1 0 0 ) ( 0 0 2 ) Eo = H(o) ( 0 2 0 ) where Eo is a constant with units of energy. (a) Find the eigenvalues and the eigenvectors. (b) Now, a perturbation ( 0 0 1 ) ( 0 1 0 ) ε = H(1) ( 1 0 0 ) is also present so that the total Hamiltonian is H(o) + H(1). Find the first order change in all eigenvalues. (c) Repeat for the second order changes.
Consider an electron in a three dimensional harmonic oscillator potential with potential V(r) = kr2/2 . A magnetic field in the z-direction is also present. The magnetic field couples to the magnetic moments due to both the orbital and the spin angular momenta. Find the first order change in the energy of the second excited state of the system.
A one-dimensional potential well has the potential { -Vo (1 - |x/a|) for |x| < a V(x) = { 0 for a < |x| < 2a { ∞ for |x| > 2a Estimate the energy values for the ground and the first excited states for a particle in this potential using the WKB method.
The spin of a 1/2-spin particle interacts with the magnetic field through the Hamiltonian H = -μ⋅B where μ is the magnetic moment due to spin and B is the magnetic field. Initially, the magnetic field is constant in the z-direction (B=zBo), and the spin is in its ground state. At time t=0 an additional perturbing magnetic field in the x-direction appears: { 0 for t<0 and t>τ Bx(t) = { { B'cos(Ωt) for 0<t<τ Find the probability that the spin is in the excited state when t>τ.
Each homework is to be placed as a single PDF document (which may contain photos of your work) and submitted through moodle.
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Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade. (The grading assistants have authority in setting up their own deadlines for submitting late homework. In any case, all late homework must be submitted before the last week of classes.)