0198506961.pdf

28 The hydrogen atom

Table 2.2 Radial hydrogenic wavefunctionsRn,lin terms of the variableρ =

Zr/(na 0 ), which gives a scaling that varies withn. The Bohr radiusa 0 is defined in

eqn 1.40.

R 1 , 0 =

(

Z

a 0

) 3 / 2

2e−ρ

R 2 , 0 =

(

Z

2 a 0

) 3 / 2

2(1−ρ)e−ρ

R 2 , 1 =

(

Z

2 a 0

) 3 / 2

2

√

3

ρe−ρ

R 3 , 0 =

(

Z

3 a 0

) 3 / 2

2

(

1 − 2 ρ+

2

3

ρ^2

)

e−ρ

R 3 , 1 =

(

Z

3 a 0

) 3 / 2

4

√

2

3

ρ

(

1 −

1

2

ρ

)

e−ρ

R 3 , 2 =

(

Z

3 a 0

) 3 / 2

2

√

2

3

√

5

ρ^2 e−ρ

Normalisation:

∫∞

0

R^2 n,lr^2 dr=1

These show a general a feature of hydrogenic wavefunctions, namely

that the radial functions forl= 0 have a finite value at the origin, i.e.

the power series inρstarts at the zeroth power. Thus electrons with

l= 0 (called s-electrons) have a finite probability of being found at the

position of the nucleus and this has important consequences in atomic

physics.

Inserting|E|from eqn 2.20 into eqn 2.17 gives the scaled coordinate

ρ=

Z

n

r

a 0

, (2.21)

where the atomic number has been incorporated by the replacement

e^2 / 4 π 0 →Ze^2 / 4 π 0 (as in Chapter 1). There are some important prop-

erties of the radial wavefunctions that require a general form of the

solution and for future reference we state these results. The probability

density of electrons withl= 0 at the origin is

|ψn,l=0(0)|^2 =

1

π

(

Z

na 0

) 3

. (2.22)

For electrons withl
= 0 the expectation value of 1/r^3 is

〈

1

r^3

〉

=

∫∞

0

1

r^3

Rn,l^2 (r)r^2 dr=

1

l

(

l+^12

)

(l+1)

(

Z

na 0

) 3

. (2.23)

These results have been written in a form that is easy to remember;

they must both depend on 1/a^30 in order to have the correct dimensions

and the dependence onZfollows from the scaling of the Schr ̈odinger