BioPHYSICAL chemistry

with a factor of iZ. Since the wavefunctions usually involve more than
one dimension, for example, x, y, and zfor a three-dimensional problem,
the derivatives are written as partial derivatives rather than full derivatives.
To show a simple derivation of Schrödinger’s equation, these operators
are substituted into the classical expression for energy, with each side of
the equation multiplied by the wavefunction ψ(x,t):

(9.25)

(9.26)

(9.27)

In dealing with the hydrogen atom, we will need to expand the equation
from a one-dimensional problem to a three-dimensional problem. By
writing the equation in terms of three positional coordinates, the most
general form of Schrödinger’s equation is obtained:

(9.28)

where ∇^2 represents the second derivative with respect to all of the spatial
variables. The three-dimensional positional vector Jcan be expressed in
Cartesian coordinates, (x,y,z), or spherical coordinates, (r,θ,φ) (Figure 9.6),
that are related according to:

i
t

rt

m

Z rt Vr rt

∂ Z

∂

ψψψ(,)=− ∇ (,)+ () (,)

2

2

2

i
t

xt

mx

Z xt V x xt

∂ Z

∂

ψ

∂

∂

(,)=− ψψ(,)+ () (,)

22

2 2

i

t

xt

m

i

x

ZZ

∂

∂

ψ

∂

∂

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ =−

⎛

⎝

⎜⎜

⎞

⎠

(,) ⎟⎟ +

1

2

2

VVx() (,)xt

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ψ

E

p

m

=+Vx()

2

2

CHAPTER 9 QUANTUM THEORY 185

Table 9.1

Physical variables and the corresponding quantum operators.

Variable Operator

Xx

VV

px

tt

E i

t

Z

∂

∂

−i

x

Z

∂

∂