BioPHYSICAL chemistry

determine the total spin and zprojection of spin accord-

ing to the relationships:

(12.40)

Electrons, protons, and neutrons all have a fixed value s

of 1/2, yielding a total spin of. The allowed values

of the zprojection are restricted to msequal to+1/2 and

−1/2, corresponding to spin values of ±Z/2. Despite its

name, this property does not represent a spinning motion

of the electrons and, despite the differences in the masses

of electrons and protons, their spin values are equal. This

quantization of these parameters is often represented by use of diagrams,

as shown in Figure 12.12.

3

2

Z^2

Smzsψψ ψ==±Z

Z

2

Sss^22

2

1

3

2

ψψψ=+()Z =

Z

258 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

ms   (^12)
ms   (^12)
Figure 12.12Particles with a value of
s=1/2 can have two spin projections
along a direction, ms=+1/2 or up,
and ms=−1/2 or down.

Derivation box 12.2 Relativistic equations

Because its value is fixed, spin must represent an intrinsic property of the electron rather than

a property that can change (such as the orbital angular momentum). Schrödinger’s equation

does not deal with this property because there is no true analogy with the properties of

particles in classical physics. Rather, it comes from relativistic corrections to classical physics.

Schrödinger’s equation is derived from the classical law of mechanics defining the total energy

Ein terms of the kinetic and potential energy (Chapter 9):

(db12.43)

The operators for momentum, p, potential, V, and energy, E(Table 9.1) are:

(db12.44)

Substitution of these operators into eqn db12.43 leads to the time-dependent form of

Schrödinger’s equation (eqn 9.28):

(db12.45)

1

2

2

mi

Vi

t

Z

∇ Z

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ ψψ+=

∂

∂

ψ

p

i

VV Ei

t

→∇ → →

Z

and Z

∂

∂

p

m

VE

2

2

+=