BioPHYSICAL chemistry

CHAPTER 12 THE HYDROGEN ATOM 259

which can be written for the time-independent case (eqn 9.39) as:

(db12.46)

The classical laws are not valid for very fast motion, where relativity becomes important. As
with quantum mechanics, the relativistic equations supplement the classical laws and agree
with the classical laws in certain limits. The corresponding relativistic equation is:

c^2 p^2 +m^2 c^4 =E^2 (db12.47)

This equation reduces to the familiar E=mc^2 for a particle at rest (and p=0). Substitution
of the operators for momentum and energy (eqn db12.44) into the relativistic expression
for energy (eqn db12.47) leads to a relativistic formulation of quantum mechanics as
expressed by:

(db12.48)

This equation is known as the Klein–Gordon equation and unfortunately is not useful for
electrons. It does, however, hold for a class of particles known as bosons that have integer
spin. An example of a boson is the photon.
The equation can be revised for use with the other class of particles, fermions, that have
half-integer spin, including the electron. The correct Fermi–Dirac equation looks like a
linearized version of the Klein–Gordon equation:

(db12.49)

In this equation αand βare 4 ×4 matrices and the wavefunctions must then be vectors
with four components. If the components are paired in up and down states, electrons of
necessity have two new characteristics. Two of these components arise from the spin, which
can be up or down. The second set arises from two states of energy, the normal-energy state
and the negative-energy state, or, as we would say today, particles and antiparticles. Thus,
there are four states: spin up and spin down for electrons and spin up and spin down for
positrons.
Positrons are the antiparticle version of electrons and are the subject of many science
fiction novels. They are not present normally but can exist in synchrotron experiments
where energy is converted into an electron and positron pair for very short periods of time.
In such cases, both the particle and antiparticle are created in equal balance to maintain the
overall charge neutrality. The minimal input energy needed to create the pair is equal to

αψβψ

∂

∂

ci mc i ψ

t

()ZZ∇+^2 =

−∇+ =

Z^22

2 m

ψψψVE

−∇+ZZ^222242 =−

2

cmcψψt 2

∂

∂

ψ

c
i

mc i

t

2

2

24

2

Z

∇ Z

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ +=

⎛

⎝

⎜⎜

⎞

⎠

ψψ⎟⎟

∂

∂

ψ