- Spins In A Magnetic Field Suppose We Begin With An Electron Having Spin Magnetic Moment Us Ys Where Y Is The Gyroma 1 (131.49 KiB) Viewed 44 times
- Spins In A Magnetic Field Suppose We Begin With An Electron Having Spin Magnetic Moment Us Ys Where Y Is The Gyroma 2 (96.77 KiB) Viewed 44 times
- Spins in a Magnetic Field Suppose we begin with an electron having spin magnetic moment us = ys, where y is the gyromagnetic ratio of the electron and Š is its spin vector. The Hamiltonian of the electron Š under an external magnetic field B becomes: H = -1, B = =YS .B a) Suppose B is pointing along the z-axis: B = 0& + 09 + Boî, where B, is the magnitude of this field. Then, only the z-component of spin S, remains in the Hamiltonian. Write down the Hamiltonian for this setup in matrix form (see Chapter 4.4.2): = À = b) Find the eigenvalues and eigenstates of the above Hamiltonian.
Suppose the electron is measured to be at the spin-up state: 14p >= 11>= ($). Also, assume that a small magnetic field is applied in the x-direction: B' = Bxł. i) Treating B' as a perturbation, find the perturbing Hamiltonian H' in matrix form. H' = ii) Show that the first order change in energy E(1) due to this perturbation is zero. ii) The two possible eigenstates of this system are spin-up 14, >=11>= [d] and spin- down 142 >=11>= 1). Calculate the second order change in energy E(2) of (42) due to the perturbation by evaluating the summation below, and using the eigenvalues calculated in part (b): (2 Ε) - <4:|H'V, >12 E1 - E i #1