Question

1. (50 pts) Consider the spin degree of freedom of an electron under an external magnetic field in the r-direction. The spin is initially (at time t 0) in the z-direction. (a) Write down the Hamiltonian for the electron spin. (Do you remember the elementary magnetic moment of electron, the Bohr magneton μΒ, in terms of electron mass me and the elementary charge e?) b) Write down the Schrödinger equation for the spin (c) Describe the motion of spin direction

Answer 1

"Larmer precession: - An electron moving around the nucleus of an atom is equivalent to magnetic dipole, So when the atom is placed external magnetic field the electron orbit processes about field direction as axis. The precession is called ""Lamar precession"". i) So Hamiltonian will be described E = -u vector middot B vector = e/2m g_T J vector middot B vector E = mu_B B g_T mg where -j lessthanorequalto M_j lessthanorequalto j N_mj = 2j + 1 cath energy level will be split into (2J + 1) energy SO t can be written in form of H H rightwardsdoublearrow mu_B B g_s mJ mu_B = e h/2m H rightarrow e/2m g_T J bar middot B bar H rightwardsdoublearrow mu_B g_T middot B vector_x H rightwardsdoublearrow mu_B B bar_x B_x rightwardsdoublearrow - mu_B (h/2) B_x H vert t > mu_B (h/2) B_x g_T rightwardsdoublearrow 1 S_n rightwardsdoublearrow h/2 S_n rightwardsdoublearrow -h/2 (2) eigen values & eigen vectors Actually direction of magnetic field (B) along direction so spin will take S_x rightwardsdoublearrow ? "