Introduction to Quantum Mechanics · Problem 8.23

The Variational Principle — David J. Griffiths

Physics — Quantum Mechanics · variational_spin_magnetic_field

Problem

As an explicit example of the method developed in Problem 8.22, consider an electron at rest in a uniform magnetic field

\mathbf{B} = B_z\hat{k}

, for which the Hamiltonian is (Equation 4.158):

H^0 = \frac{eB_z}{m}S_z. \qquad (8.76)

The eigenspinors,

\chi_a

and

\chi_b

, and the corresponding energies,

E_a

and

E_b

, are given in Equation 4.161. Now we turn on a perturbation, in the form of a uniform field in the

x

direction:

H' = \frac{eB_x}{m}S_x. \qquad (8.77)

(a) Find the matrix elements of

H'

, and confirm that they have the structure of Equation 8.74. What is

h

? (b) Using your result in Problem 8.22(b), find the new ground state energy, in second-order perturbation theory. (c) Using your result in Problem 8.22(c), find the variational principle bound on the ground state energy.

Related theory

Magnetic Force And Magnetic Field

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