Physical Chemistry Third Edition

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. Aneven functionhas the property thatf(−x)

f(x), and anodd functionhas the property thatf(−x)−f(x). Equation (15.2-19)

makes it clear that we have an oscillatory solution. If the factorihad 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

Ĥ−h ̄

2

2 m

(

∂^2

∂x^2

+

∂^2

∂y^2

+

∂^2

∂z^2

)

+V(x,y,z)−

h ̄^2

2 m

∇^2 +V(x,y,z) (15.2-25)

The operator∇^2 is called theLaplacian 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.