Page 169 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

160 The hydrogen atom

6.2 The hydrogen-like atom
È
The Schrodinger equation (6.12) for the relative motion of a two-particle
system is applicable to the hydrogen-like atom, which consists of a nucleus of
charge ‡Ze and an electron of charge ÿe. The differential equation applies to
H for Z ˆ 1, He for Z ˆ 2, Li 2‡ for Z ˆ 3, and so forth. The potential energy
‡
V(r) of the interaction between the nucleus and the electron is a function of
2 1=2
2
2
their separation distance r ˆjrjˆ (x ‡ y ‡ z ) and is given by Coulomb's
law (equation (5.76)), which in SI units is
Ze 2
V(r) ˆÿ
4ðå 0 r
where meter is the unit of length, joule is the unit of energy, coulomb is the
unit of charge, and å 0 is the permittivity of free space. Another system of units,
used often in the older literature and occasionally in recent literature, is the
CGS gaussian system, in which Coulomb's law is written as
Ze 2
V(r) ˆÿ
r
In this system, centimeter is the unit of length, erg is the unit of energy, and
statcoulomb (also called the electrostatic unit or esu) is the unit of charge. In
this book we accommodate both systems of units and write Coulomb's law in
the form
Ze9 2
V(r) ˆÿ (6:13)
r
where e9 ˆ e for CGS units or e9 ˆ e=(4ðå 0 ) 1=2 for SI units.
Equation (6.12) cannot be solved analytically when expressed in the
cartesian coordinates x, y, z, but can be solved when expressed in spherical
polar coordinates r, è, j, by means of the transformation equations (5.29). The
2
laplacian operator = in spherical polar coordinates is given by equation (A.61)
r
and may be obtained by substituting equations (5.30) into (6.9b) to yield
" #

1 @ @ 1 1 @ @ 1 @ 2
2 2
= ˆ r ‡ sin è ‡
r
2
2
r @r @r r 2 sin è @è @è sin è @j 2
If this expression is compared with equation (5.32), we see that

1 @ @ 1
2 2 ^ 2
= ˆ r ÿ L
r 2 2 2
r @r @r " r
^ 2
where L is the square of the orbital angular momentum operator. With the
2
laplacian operator = expressed in spherical polar coordinates, the SchroÈdinger
r
equation (6.12) becomes