Physical Chemistry , 1st ed.

In the case of the hydrogen atom, the potential energy is not zero. There is
an interaction between the electron and the nucleus for this system. The in-
teraction is electrostatic; that is, it is due to the attraction between the posi-
tively charged nucleus and the negatively charged electron. Fortunately, such
an electrostatic potential energy has a known mathematical formula, based ul-
timately on the ideas of Coulomb:

V

4





e^2

0 r

 (11.57)

where e1.602
10 ^19 coulombs, 0 is the permittivity of free space and
equals 8.854
10 ^12 C^2 /J m, and ris the distance between the two charged
particles. From this, it is easy to show that the expression for Vin equation
11.57 yields units of J, a unit of energy.
The potential energy Vdepends only on the distance rseparating the nu-
cleus and the electron, not the angles or . This means that the potential
energy is the same at a fixed rno matter what the values ofor are. That is
to say, the potential energy is spherically symmetric.The force between the elec-
tron and the hydrogen nucleus is also spherically symmetric. Because of this,
the force is said to be a central force,and the hydrogen atom description in
quantum mechanics is an example of what is generally known as a central force
problem.
The complete Schrödinger equation for this central force problem is thus

 2





2



r

1

2 



r

r^2 





r



r^2 s

1

in









sin 











r^2 s

1

in^2 









2

 (^2)  4





e

 0

2

r

 E (11.58)

and acceptable wavefunctions for a hydrogen atom must satisfy this Schrödinger
equation. It should be noted that there is another way to write this Schrödinger

equation, using the total angular momentum operator Lˆ^2 :

 2





2



r

1

2 



r



r

(^2) 




r



2 

1

r^2

ˆL^2 

4





e

 0

2

r

E (11.59)

11.10 The Hydrogen Atom: The Quantum-Mechanical Solution

A detailed mathematical solution of equations 11.58 or 11.59 is not presented
here, but the approach will be explained. As with the 3-D rotational motion, it
will be assumed that acceptable wavefunctions are separable into three func-
tions that depend only on r,on , and on :

(r,,) R(r) () ()

It may not be surprising to learn that the and parts of the wavefunction
are the spherical harmonics, discussed earlier for the 3-D rigid rotor. These
solutions impose two integers called quantum numbers,and m, which de-
termine the exact mathematical expression. Because the Schrödinger equation

can be written in terms of the total angular momentum operator Lˆ^2 , we can

substitute the solutions for that part of the operator into the Schrödinger equa-
tion and get a differential equation in terms ofrand Ralone:





2 

^2



r

1

2 



r

r^2 





r



^2 

2

(

r



2

1)



4





e^2

0 r

RER (11.60)

11.10 The Hydrogen Atom: The Quantum-Mechanical Solution 353