Introduction to Quantum Mechanics · Problem 7.51

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · direct_perturbation_solving

Problem

Sometimes it is possible to solve Equation 7.10 directly, without having to expand

\psi_n^1

in terms of the unperturbed wave functions (Equation 7.11). Here are two particularly nice examples. (a) Stark effect in the ground state of hydrogen. (i) Find the first-order correction to the ground state of hydrogen in the presence of a uniform external electric field

E_{\text{ext}}

(see Problem 7.45). *Hint:* Try a solution of the form

\left(A + Br + Cr^2\right)e^{-r/a}\cos\theta;

your problem is to find the constants

A

B

, and

C

that solve Equation 7.10. (ii) Use Equation 7.14 to determine the second-order correction to the ground state energy (the first-order correction is zero, as you found in Problem 7.45(a)). *Answer:*

-m\left(3a^2 eE_{\text{ext}}/2\hbar\right)^2

. (b) If the proton had an electric dipole moment

p

, the potential energy of the electron in hydrogen would be perturbed in the amount

H' = -\frac{ep\cos\theta}{4\pi\epsilon_0 r^2}.

(i) Solve Equation 7.10 for the first-order correction to the ground state wave function. (ii) Show that the *total* electric dipole moment of the atom is (surprisingly) *zero*, to this order. (iii) Use Equation 7.14 to determine the second-order correction to the ground state energy. What is the first-order correction?

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