) cos(0/2) + -2) state is placed in a magnetic field with strength B pointing 4....
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...
3) An electron whose spin is quatized in n = (sin (1) cos (0), sin (1) sin (0), cos (7)] is in a magnetic field B = (0,0, Bo). Use the Hamiltonian H= -7B.ő (a) Calculate the time development of the spin orientation by calculating the expectation value (sx). What is the period T after which the spin returns (the first time) into its initial orientation. (b) What is the energy during this precession and the resulting (energy caused) phase...
An electron of magnetic moment υ = γ S is placed in a magnetic field B = (-ωx/γ, -ωy/γ, -ωz/γ). Show that the evolution operator of the spin is U(t,0) = exp(-iυt)
An electron moves with speed 5.0 times 10^5m/s in a uniform magnetic field of strength 0.60 T that points Hast. At some instant, the electron experiences an upward magnetic force of magnitude 4.0 times 10^14 N. In what direction is the electron moving at that instant? [If there is more than one possible direction, describe all the possibilities.] On a carefully drawn and labeled diagram, show the (possible) direction(s) of the electron's velocity relative to the magnetic field and the...
6. Given the spin Hamiltonian of an electron in a magnetic field B-Bok, H--y5.B, find the time evolution Unitary operator given as U(6) - exp(-()/N). Given that the initial quantum state is le (O) >= 0) find the state after a time t given as (10 points) () >= U(t)|(0) >
9. An electron moving with non-relativistic velocity v in an electric field E experiences a magnetic fieldB given by: v x (-V(r)) v x E B=- where (r) is the electric potential. This magnetic field interacts with the magnetic moment u of the electron given by -S, =n me where S is the electron spin. Assuming non-relativistic mechanics, show that the Hamiltonian representing this effect (spin-orbit coupling) for a spherically-symmetric electric potential is: 1 dφ(r) S.L ΔΗ [6] r dr...
4.8. A spin- particle, initially in a state with S h/2 with n sin i+ cos k, is in a constant magnetic field Bo in the z direction. Determine the state of the particle at time and determine how (S,), (S), and (S.) vary with time.
What should be the strength of the magnetic field in T (2 significant figures) if the Larmor frequency for protons in this magnetic field is 150 MHz?
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...
Q7M.6 Assume that the 1+z) and -z) states for an elec- tron in a magnetic field are energy eigenvectors with ener- gies E and 0, respectively, and assume that the electron's state at t=0 is 14(0) = (Q7.25) Find the probability that we will determine this electron's spin to be in the +x direction at time t.