The behavior of a spin- particle in a uniform magnetic field in the z-direction, , with the Hamiltonian
You found that the expectation value of the spin vector undergoes Larmor precession about the z axis. In this sense, we can view it as an analogue to a rotating coin, choosing the eigenstate with eigenvalue to represent heads and the eigenstate with eigenvalue to represent tails. Under time-evolution in the magnetic field, these eigenstates will “rotate” between each other.
(a) Suppose at , we measure to be . Find the probability that a measurement of will again yield at time . (Where applicable, write your answers for all parts of this problem using the Larmor frequency, .)
(b) For small , what is the probability, to leading order in , that the measurement will instead yield ? What is meant by “small” here (i.e., small compared to what)?
Hint: you may find it useful to recall that the Taylor expansion of to leading order in x is given by:
The behavior of a spin- particle in a uniform magnetic field in the z-direction, , with...
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...
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...
The energy of a magnetic moment in a magnetic field is . A certain paramagnetic salt contains 1025 magnetic moments per m3. Each one has a value , due to the atom's spin. As the spin is 1/2, there only are two possible states and the magnetic moments can be parallel or antiparallel to the field. Each magnetic moment belongs to one distinguishable atom. A 1 cm3 sample of this salt is placed in a electromagnet producing a uniform magnetic...
Intro to Quantum Mechanics Problem: An electron under the influence of a uniform magnetic field By in the y-direction has its spin initially (at 0) pointing in the positive x-direction. That is, it is in an eigenstate of S with eigenvalue +,S h. The Hamiltonian H--μ . B-γ By Sy consists of the interaction of the magnetic dipole moment μ due to spin and the magnetic field B. Show that the probability of finding the electron with its spin pointing...
Consider an electron in a uniform magnetic field along the z direction. A measurement shows that the spin is along the negative x direction at -0. a. Find the eigenvector describing the initial spin state. 5. 0 -1 b. Write the Hamiltonian as a 2x2 matrix by starting with H =-7S-Band taking the field B in the z- direction. Find the energy eigenvalues and eigenvectors. Solve for | Ψ(t) using these eigenvalues, eigenvectors, and the initial condition from part a....
) cos(0/2) + -2) state is placed in a magnetic field with strength B pointing 4. Larmor precession: an electron prepared in the V(t 0 sin(0/2)e in the a-direction. Calculate the time evolution of the electron's spin state. In addition calculate the time evolution of (S), S and (S ). (2 points)
Consider the state of a spin-1/2 particle 14) = v1o (31+z) + i] – z)) where | z) are the eigenstates of the operator of the spin z-component $z. 1. Show that [V) is properly normalized, i.e. (W14) = 1. 2. Calculate the probability that a measurement of $x = 6x yields 3. Calculate the expectation value (Šx) for the state 14) and its dispersion ASx = V(@z) – ($()2. 4. Assume that the spin is placed in the magnetic...
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
Calculate the magnetic field, i.e. the magnetic flux density, B, at the center of they hydrogen atom for the state and for the . Further, estimate the magnitude of the orbital magnetic field experienced by a 2p-electron in hydrogen. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
(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...