Question

Problem 7.49

Problem 7.49 A hydrogen atom is placed in a uniform magnetic field Bo Bo (the Hamiltonian can be written as in Equation 4.230). Use the Feynman-Hellman theorem (Problem 7.38) to show that a En (7.114) where the electron's magnetic dipole moment10 (orbital plus spin) is Yo l-mechanical + γ S . μ The mechanical angular momentum is defined in Equation 4.231 a volume V and at 0 K (when they're all in the ground state) is41 Note: From Equation 7.114 it follows that the magnetic susceptibility of N atoms in (7.115) where Eo is the ground-state energy. Although we derived Equation 7.114 for a hydrogen atom, the expression applies to multi-electron atoms as well-even when electron-electron interactions are included. Problem 7.38 Suppose the Hamiltonian H, for a particular quantum system, is a function of some parameter λ; let En (A) and ψη(A) be the eigenvalues and eigenfunctions of Ha). The Feynman-Hellmann theorem 35 states that ψη (7.110) (assuming either that En is nondegenerate, or-if degenerate-that the Vns are the "good" linear combinations of the degenerate eigenfunctions). (a) Prove the Feynman-Hellmann theorem. Hint: Use Equation 7.9 (b) Apply it to the one-dimensional harmonic oscillator, (i) using A- (this yields a formula for the expectation value of V), (ii) usingA h (this yields (T), and (i) using à m (this yields a relation between (T) and (V)). Compare your answers to Problem 2.12, and the virial theorem predictions (Problem 3.37). (b) Show that the Hamiltonian can be written p2 8 m

Answer 1

Pa ge N DEn 8 m rrn 80 OEo B6