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4.8 The Hydrogen Atom

This is an electron moving around a proton. Let m be the mass of the electron, e its
charge, and h Planck’s constant divided by 2π. Let the origin of coordinates (x, y, z) be
2
2 1/2
2
at the proton and let r = (x + y + z ) be the spherical coordinate. Then the motion
of the electron is given by a ”wave function” u(x, y, z, t) which satisfies Schr¨odinger’s
equation
h 2 e 2
−ihu t = ∆u + u
2m r
RRR
2
in all of space −∞ < x, y, z < +∞. We are supposed to have |u| dxdydz = 1
√
(integral over all space). Note that i = −1 and u is complex-valued. The coefficient
2
function e /r is called the potential. For any other atom with a single electron, such as
2
2
a helium ion, e is replaced by Ze , where Z is the atomic number.
What does this mean physically? In quantum mechanics quantities cannot be measured
exactly but only with a certain probability. The wave function u(x, y, z, t) represents a
possible state of the electron. If D is any region in xyz-space, then
ZZZ
2
|u| dxdydz
D
is the probability of finding the electron in the region D at the time t. The expected z
coordinate of the position of the electron at the time t is the value of the integral
ZZZ
2
z|u(x, y, z, t)| dxdydz;
D
similarly for the x and y coordinates. The expected z coordinate of the momentum is
ZZZ
∂u
−ih (x, y, z, t) · u(x, y, z, t)dxdydz,
D ∂z
where u is the complex conjugate of u. All other observable quantities are given by
operators A, which act on functions. The expected value of the observable A equals
ZZZ
. Au(x, y, z, t) · u(x, y, z, t)dxdydz.
D
Thus the position is given by the operator Au = xu and the momentum is given by the
operator Au = −ih∇u.
Schr¨odinger’s equation is most easily regarded simply as an axiom that leads to the
correct physical conclusions, rather than as an equation that can be derived from simpler
principles. It explains why atoms are stable and don’t collapse. It explains the energy
levels of the electron in the hydrogen atom observed by Bohr. In principle, elaborations
of it explain the structure of all atoms and molecules and so all of chemistry! With
many particles, the wave function u depends on time t and all the coordinates of all the
particles and so is a function of a large number of variables. The Schr¨odinger equation
then becomes
n
X h 2
−ihu t = (u x i x i + u y i y i + u z i z i ) + V (x 1 , . . . , z n )u
2m i
i=1

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