Introduction to Quantum Mechanics · Problem 4.15

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · hydrogen_expectation_values

Problem

(a) Find

\langle r \rangle

and

\langle r^2 \rangle

for an electron in the ground state of hydrogen. Express your answers in terms of the Bohr radius. (b) Find

\langle x \rangle

and

\langle x^2 \rangle

for an electron in the ground state of hydrogen. Hint: This requires no new integration—note that

r^2 = x^2 + y^2 + z^2

, and exploit the symmetry of the ground state. (c) Find

\langle x^2 \rangle

in the state

n = 2

\ell = 1

m = 1

. Hint: this state is not symmetrical in

x

y

z

. Use

x = r \sin\theta \cos\phi