Concise Physical Chemistry

c16 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

252 WAVE MECHANICS OF SIMPLE SYSTEMS

One of the postulates of quantum mechanics is as follows:

If a system is in a statesdescribed by|〉an eigenvector of the operatorAˆ (or,

equivalently ifis an eigenfunction ofAˆ), the correspondingobservable ais an

eigenvalue:

Aˆ|〉=a|〉

or, equivalently,

Aˆ=a

and experiments onsin state|〉willalwaysyield the observablea.

In the case of the atomic and molecular systems that concern us here, the general

operatorAˆis the Hamiltonian operator,Hˆ, and the observableais the energy level

Eiof the state|i〉. Usually there are many states leading to many energiesEi.

The set{Ei}represents aspectrumor multiplicity of energy levels, one for each

eigenvector:

Hˆ|i〉=Ei|i〉

An operator operating on a vector produces another vector. The operatorHˆ op-

erating on|〉produces the vector

∣

∣Hˆ|〉. The inner product of〈|premultiplied

into

∣

∣Hˆ|〉is

〈|Hˆ|〉=〈|E|〉=E〈|〉

The inner product of two vectors〈|〉is a scalar, which is why the eigenvalueE

can be moved out of the brackets on the right. TheoperatorHˆ cannot be factored

out. This leads to an expression for the energy

E=

〈|Hˆ|〉

〈|〉

The Born probability postulate leads to a further simplification. The inner product

〈|〉amounts to the integral over all space of the wave function squared^2 or

the product∗if the wave function is complex. The sum of the probabilities of

finding an electron over all space must be 1.0 (certainty) because the electron has

to be somewhere. With the inner product〈|〉= 1 .0 we have the energy of an

eigenstate as

Ei=〈i|Hˆ|i〉