28 Chapter 1Numbers, variables, and units
In this form the equation is often referred to as the ‘Schrödinger equation in atomic
units’. The results of computations are then numbers that must be reinterpreted
as physical quantities. For example, the quantity Ein equation (1.18) is an energy.
Solution of equation (1.19) gives the numbersE 1 = 1 − 122 n
2, for all positive integers n,
and these numbers are then interpreted as the energiesE 1 = 1 − 122 n
2E
h.
EXAMPLE 1.21The atomic unit of energy
By Coulomb’s law, the potential energy of interaction of chargesq
1andq
2separated
by distance ris
whereε
01 = 1 8.85419 1 × 110
− 12F m
− 1is the permittivity of a vacuum. For chargesq
11 = 1 Z
1e
andq
21 = 1 Z
2eseparated by distancer 1 = 1 Ra
0,
(i) To show that the unit is the hartree unitE
hin Table 1.4, usea
01 = 14 πε
0A
22 m
ee
2:
(ii) To calculate the value of E
hin SI units, use the values of eand a
0given in
Table 1.4. Then
= 1 (4.35975 1 × 110
− 3) 1 × 1 (10
− 15) 1 × 1 (C
21 F
− 1)
From the definitions of the coulomb C and farad F in Table 1.2,F 1 = 1 C
21 J
− 1so that
C
21 F
− 11 = 1 J. Therefore
0 Exercise 108
e
a
E
200184
10
πε
=×=
−4.35975 J
he
a
20024
1 60218
πε 4 3 14159 8 85419 5 29177
=
×××
.
...
×
×
×
×
−−−−10 10
10 10
19 1912 112C
F m m
−
1e
a
e
me
2002002244
4
ππ
π
εε
ε
=
÷
=
eee
me me
20202424
416
π
ππ
ε
ε
×
=
eeεε
022=E
hV
ZZ
R
e
a
=
122004 πε
V
r
=
1204 πε