Introduction to Quantum Mechanics · Problem 4.55

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · hydrogen_radial_wave

Problem

(a) Use the recursion formula (Equation 4.76) to confirm that when

\ell = n - 1

the radial wave function takes the form

R_{n(n-1)} = N_n r^{n-1} e^{-r/na}

, and determine the normalization constant

N_n

by direct integration. (b) Calculate

\langle r\rangle

and

\langle r^2\rangle

for states of the form

\psi_{n(n-1)m}

. (c) Show that the "uncertainty" in

r

(

\sigma_r

) is

\langle r\rangle / \sqrt{2n+1}

for such states. Note that the fractional spread in

r

decreases, with increasing

n

(in this sense the system "begins to look classical," with identifiable circular "orbits," for large

n

). Sketch the radial wave functions for several values of

n

, to illustrate this point.