Problem 2. (20 points) Consider a spin-1/2 particle with gyromagnetic ratio γ in a magnetic field B- Bi+B,k . (a) Treat...
qm 2019.3 3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with charge e and mass m in a magnetic field B is À eB B. Ŝ, m where Ŝ are the spin angular momentum operators. You should make use of expres- sions for the spin operators that are given at the end of the question. (i) Write down the energy eigenvalue equation for this particle in a field directed along the y axis, i.e. B...
4. (20 points). δ. unction perturbation. Consider a particle in a one-dimensional infinite square well with boundaries at -a and-a. We introduce the following 6-function perturbation at x=0: a. Compute the first-order corrections to the energies of the particle induced by the ν' perturbation b. Recall that you solved this problem exactly in problem set 4 (Griffiths 2.43). Compare your perturbation theory result to the exact solution.
(2.1) (20 points) A spin 1/2 particle is in an eigenstate of Sy with eigenvalue h/2 at the initial time t = 0. At that time, it is placed in a magnetic induction B = B2, and it is then allowed to precess in that induction for the time T. Then, at that instant T, B is instantaneously rotated from the z to the y direction, becoming B = Bį. After another identical time interval T occurs, a measurement of...
4. (20 points). 5-function perturbation. Consider a particle in a one-dimensional infinite square well with boundaries at x--a and x-a. We introduce the following δ-function perturbation at V'(x) 00(z). a. Compute the first-order corrections to the energies of the particle induced by the perturbation b. Recall that you solved this problem exactly in problem set 4 (Griffiths 2.43). Compare your perturbation theory result to the exact solution
2. (25 points). Rabi oscillations. Consider a spin-1/2 particle in a magnetic field B - Bo2 such that the spin eigenstates are split in energy by hwo (let's label the ground state |0) and the excited state |1)). The Hamiltonian for the system is written as hwo Zeeman - _ here and below. ơng,z are the usual Pauli matrices. A second, oscillating field is applied in the transverse plane, giving rise to a time-dependent term in the Hamiltoniain hw Rabi-...
Problem 111.3. A spin 1/2 particle interacts with a nnagnetic field B = Boe through the Pauli interaction H-μσ. B where μ is the magnetic moment. The Pauli spin matrices are İ-(Oz,@yMwwhere the σί are T0 1 0-il The eigenstates for d, are the spinors 0 (a) (3 pts.) Suppose that at time t-0 the particle is in an eigenstate Xx corresponding to spin pointing along the positive z-axis. Find the eigenstatexz in terms of α and β. (b) (7...
For a spin-1/2 particle in a magnetic field B, with energies and , (a) calculate the partition function. (b) Show that the mean energy of this particle is given by ̅ For a system of noninteracting spins, (c) what is the total partition function and (d) mean energy? We were unable to transcribe this image2 2kT 2 2kT
Problem 2. (30 points) (a) (3 points) The Stark effect (shift of energy levels by a constant external electric field) in atom is usually observed to be quadratic in the field strength. Explain why. (b) (3 points) But for some states of the hydrogen atom, the Stark effect is observed to be linear in the field strength. Explain why. (c) Ilustrate by making a perturbation calculation of the Stark effect of an electric field E Ez to lowest non-vanishing order...
Exercise 8: Time dependence of a two-level system Consider a two-level system with stationary states a and b with unperturbed energies and E() and corresponding eigenfunctions ф ) and 4°, respectively. Assume E ) > E 0. such that the Bohr angular frequency wi(EEis positive. A time-independent perturbation V is switched on at time t 0 a) Write down the coupled set of equations for the coefficients ca(t) and c(t) of the wave function of the system: Note that we...
1. Imagine a version of the particle in a box where the potential is given by: b-1 b-1 oootherwise where b is any real number greater than or equal to 2 a) Assuming that > Vo for all n, use the WKB approximation to find the energies. Give your final answer in terms of b, b, and E b) What happens in either extreme, as b approaches 2 or o°? Does the WKB approximation give the exact answers in these...