Physical Chemistry , 1st ed.

specific formula from classical mechanics. In terms of linear momentum px,

kinetic energy is given by

K

2

p

m

x

2

Schrödinger, however, thought in terms of operators acting on wavefunctions,

and so he rewrote the Hamiltonian function in terms of operators. Using the

definition of the momentum operator,

ˆpxi





x

and supposing that the potential energy is a function ofposition(that is, a

function ofx) and so can be written in terms of the position operator,

xˆx

Schrödinger substituted into the expression for the total energy to derive an

operator for energy named (for obvious reasons) the Hamiltonian operator Hˆ:

Hˆ

2



m

2





x

2

    2 Vˆ(x) (10.9)

This operator Hˆoperates on a wavefunction and the eigenvalue corre-

sponds to the total energy of the system E:

^2



m

2





x

2

    2 Vˆ(x)E (10.10)

Equation 10.10 is known as the Schrödinger equationand is a very impor-

tant equation in quantum mechanics. Although we have placed certain re-

strictions on wavefunctions (continuous, single-valued, and so on), up to now

there has been no requirement that an acceptable wavefunction satisfy any par-

ticulareigenvalue equation. However, ifis a stationary state (that is, if its

probability distribution does not depend on time), it should also satisfy the

Schrödinger equation. Also note that equation 10.10 does not include the vari-

able for time. Because of this, equation 10.10 is more specifically referred to as

the time-independent Schrödinger equation.(The time-dependentSchrödinger

equation will be discussed near the end of the chapter and represents another

postulate of quantum mechanics.)

Although the Schrödinger equation may be difficult to accept at first, it

works: when applied to ideal and even real systems, it yields the values for the

energies of the systems. For example, it correctly predicts changes in energy of

the hydrogen atom, which is a system that had been studied for decades before

Schrödinger’s work. Quantum mechanics, however, uses a new mathematical

tool—the Schrödinger equation—for predicting observable atomic phenom-

ena. Because the values of atomic and molecular observables are properly pre-

dicted by using the Schrödinger equation and wavefunctions, they are consid-

ered the proper way of thinking about atomic phenomena. The behavior of

electrons is described by a wavefunction. The wavefunction is used to deter-

mine all properties of the electrons. Values of these properties can be predicted

by operating on the wavefunction with the appropriate operator. The appro-

priate operator for predicting the energy of the electron is the Hamiltonian op-

erator.

To see how the Schrödinger equation works, the following example illus-

trates how the Hamiltonian operates on a wavefunction.

286 CHAPTER 10 Introduction to Quantum Mechanics