15.2 The Schrödinger Equation
661
In the final version of Eq. (15.2-19) we have used the fact that the cosine is an even
function and the sine is an odd function. An even function has the property that f (−x)
f (x), and an odd function has the property that f (−x) −f (x). Equation (15.2-19)
makes it clear that we have an oscillatory solution. If the factor i had been omitted from
the time-dependent Schrödinger equation, the time-dependent factor in the solution
would have been
η(t) e
−Et/h (not usable)
(15.2-20)
which is clearly not oscillatory. The real part and the imaginary part oscillate with the
same frequency, but out of phase (with their maximum values at different times).
The real and imaginary parts of the complete wave function have stationary nodes in
the same locations, since they combine with the same coordinate factor.
If we write Eq. (15.2-19) in the form
η(t) cos(2πEt/h) − isin(2πEt/h)
(15.2-21)
we can recognize the period as
τ
h
E
(15.2-22)
and the frequency as
ν
E
h
(15.2-23)
Notice the similarity of this relationship with the Planck–Einstein formula for the
energy of a photon, Eq. (14.4-8):
E(photon) hν
(15.2-24)
The Schrödinger Equation in Three Dimensions
For a single particle moving in three dimensions, the Hamiltonian operator is
H −
¯
h
2
2m
∂ 2
∂x 2 +
∂ 2
∂y 2 +
∂ 2
∂z 2
+ V (x, y, z) −
¯
h
2
2m
∇
2
+ V (x, y, z) (15.2-25)
The operator ∇ 2 is called the Laplacian operator, introduced in Eq. (B-45) of
Appendix B. In Cartesian coordinates
∇
2
∂
2
/∂x
2
+ ∂
2
/∂y
2
+ ∂
2
/∂z
2
(15.2-26)
In Chapter 17 we will express the Hamiltonian operator in terms of spherical polar
coordinates.