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
∂
∂