Physical Chemistry Third Edition

18.4 Excited States of the Helium Atom 773

Probability Densities for Excited States

The probability density for finding the two electrons irrespective of spins is obtained

by integrating the square of the wave function over the spin coordinates. Consider the

state corresponding toΨ 1 in Eq. (18.4-2a). The probability of finding electron 1 in

the volume elementd^3 r 1 and electron 2 in the volume elementd^3 r 2 irrespective of

spins is

(Probability)

1

2

([ψ 1 s(1)ψ 2 s(2)−ψ 2 s(1)ψ 1 s(2)])d^3 r 1 d^3 r 2

×

∫

α(1)∗α(2)∗α(1)α(2)ds(1)ds 2



1

2

[ψ 1 s(1)ψ 2 s(2)−ψ 2 s(1)ψ 1 s(2)]^2 d^3 r 1 d^3 r 2



1

2

[ψ 1 s(1)^2 ψ 2 s(2)^2 −ψ 1 s(1)ψ 2 s(2)ψ 2 s(1)ψ 1 s(2)

−(ψ 2 s(1)ψ 1 s(2)ψ 1 s(1)ψ 2 s(2)+ψ 2 s(1)^2 ψ 1 s(2)^2 ]d^3 r 1 )d^3 r 2 (18.4-3)

where we have used the fact that the 1sand 2sspace orbitals are real and the fact that

the spin functions are normalized. The probability of finding electron 1 in the volume

elementd^3 r 1 is obtained by integrating over the coordinates of particle 2:

(

Probability of finding

particle 1 ind^3 r 1

)



1

2

∫

[ψ 1 s(1)^2 ψ 2 s(2)^2 −ψ 1 s(1)ψ 2 s(2)ψ 2 s(1)ψ 1 s(2)

−(ψ 2 s(1)ψ 1 s(2)ψ 1 s(1)ψ 2 s(2)+ψ 2 s(1)^2 ψ 1 s(2)^2 ]d^3 r 2 )d^3 r 2



1

2

(ψ 1 s(1)^2 +ψ 2 s(1)^2 )d^3 r 1 (18.4-4)

The second and third terms in Eq. (18.4-3) vanish because of orthogonality when the

coordinates of particle 2 are integrated. This probability is the average of what would

occur if electron 1 occupied orbital 1 and what would occur if it occupied orbital 2,

which is reasonable since we cannot say which electron occupies which orbital. The

expression for electron 2 is the same function. For the two-electron system the total

probability of finding some electron in a volumed^3 r 1 is the sum of the probabilities

for the two electrons:

(

Probability of finding

an electron ind^3 r 1

)

(ψ 1 s(r 1 )^2 +ψ 2 s(r 1 )^2 )d^3 r 1 (18.4-5)

When this probability density is multiplied by−e, the electron charge, it is the charge

density (charge per unit volume) due to the electrons. The other three wave functions

represented in Eq. (18.4-2) give the same results.

Exercise 18.3

Show that the other three wave functions in Eq. (18.4-2) give the same result for the charge

density.