The radial function of hydrogen ( , ) is R_ 21 (r) = 1 24a_0^3

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The hydrogen atom

Problem

The

2p

radial function of hydrogen (

n=2

l=1

) is

R_{21}(r) = \frac{1}{\sqrt{24a_0^3}}\,\frac{r}{a_0}\,e^{-r/2a_0},

with

a_0 = 0.0529

nm. The proton is fixed, the electron is non-relativistic and spin is ignored. You may use

\int_0^\infty x^qe^{-x}dx = q!

for integer

q \ge 0

. ((a)) Show that

u=rR_{21}

has the two forms the chapter's asymptotic analysis requires:

u \propto r^{l+1}

r\to0

, and

e^{-\rho}

decay at large

\rho = \kappa r

with

\kappa = 1/(na_0)

; identify

n

and

l

from the power of

r

and the exponential. ((b)) Verify that

R_{21}

is normalised,

\int_0^\infty |R_{21}|^2r^2dr = 1

. ((c)) Derive the mean radius

\langle r\rangle = \int_0^\infty rP(r)\,dr

with

P(r) = |R_{21}|^2r^2

, and give it as a multiple of

a_0

and in nm.

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