Physical Chemistry , 1st ed.

Wavefunctions for hydrogen-like systems, determined by quantum num-

bers, can be labeled with those quantum numbers. Therefore, it is common to

see  1 s, 3 d, and so on.

As can be seen from Table 11.4, wavefunctions having a nonzero value for

mhave an imaginary exponential function part. This means that the overall

wavefunction is a complex function. In cases where completely real functions

are desired, it is useful to define real wavefunctions as linear combinations of

the complex wavefunctions, taking advantage of Euler’s theorem. For example:

 2 px 

1

2

( 2 p 1  2 p 1 )

(11.67)

 2 py 

i

2

( 2 p 1  2 p 1 )

The pwavefunctions defined like this are real, not complex, and so are easier

to work with in many situations. Real wavefunctions for d,f, and other orbitals

are defined similarly. These nonimaginary wavefunctions are noteigenfunc-

tions ofLˆzany longer, since they are composed of parts that have different

eigenvalues ofm. They are still eigenfunctions of the energy and total angu-

lar momentum, however. (In fact, it is only becausethe original wavefunctions

are degenerate that we are able to take linear combinations, like those in equa-

tion 11.67.)

The behavior of the wavefunctions in space raises some interesting points.

Every s-type orbital has spherical symmetry, since there is no angular depen-

dence in the wavefunction. Because the probability of an electron existing at any

point in space is related to ^2 or, in this case,R^2 , the probability of an selec-

tron existing in space is spherically symmetric also. Starting from the nucleus

and moving out along a straight line, one can plot the probability of the elec-

tron having a certain value ofrversus the radial distance ritself. Such a plot for

 1 sis shown in Figure 11.18. This plot shows the surprising conclusion that the

radius of maximum probability occurs at the nucleus,that is, where r0.

This analysis is a little misleading. From a spherical polar viewpoint, there is

very little volume of space close to the nucleus, because for all values ofand 

a small value ofrsweeps out a very tiny sphere. The total probability of the elec-

tron existing in such a small volume of space should be small. However, as the

radius increases, the spherical volume swept out by the spherically symmetric

wavefunction gets larger and larger, and one would expect an increase in proba-

bility that the electron will be located at greater distances from the nucleus.

Instead of considering the electron probability along a straight line out from

the nucleus, consider the electron probability on a spherical surface around the

nucleus, each spherical surface getting larger and larger. Mathematically, the

spherical surface corresponds not to R^2 ,but 4r^2 R^2. A plot of 4r^2 R^2 ver-

sus rfor  1 sis shown in Figure 11.19. The probability starts at zero (a conse-

quence of the “zero” volume at the nucleus), increases to a maximum, then de-

creases toward zero as the radius gets larger and approaches infinity. Quantum

mechanics shows that an electron doesn’t have a specificdistance from the nu-

cleus. Instead, it can have a range of distances having differing probabilities. It

does have a most probable distance. It can be shown mathematically that the

value ofrat the most probable distance is

rmax

4 





e

2

2

 0

 a (11.68)

a0.529 Å (11.69)

360 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom

R^2

Distance from nucleus

0

Distance from nucleus (Å)

Most probable distance  a  Bohr radius

5

4 

(^2) r
R
(^2) 
4321
Figure 11.19 A plot of 4r^2 R^2 for  1 sversus
distance from the nucleus. The 4r^2 contribution
accounts for the spherical symmetry of the 1s
wavefunction about the nucleus. By looking at
the probability of existence in spherical shells
rather than straight away from the nucleus, we
get a more realistic picture of the expected be-
havior of an electron in a hydrogen atom.
Figure 11.18 A plot of the square of the ra-
dial function of 1 sversus distance from the nu-
cleus for the hydrogen atom. It suggests that the
electron has a maximum probability of existing
at the nucleus.