Physical Chemistry , 1st ed.

10.3 Observables and Operators

When studying the state of a system, one typically makes various measurements

of its properties, such as mass, volume, position, momentum, and energy. Each

individual property is called an observable.Since quantum mechanics postulates

that the state of a system is given by a wavefunction, how does one determine

the value of various observables (say, position or momentum or energy) from

a wavefunction?

The next postulate of quantum mechanics states that in order to determine

the value of an observable, you have to perform some mathematical operation

on a wavefunction. This operation is represented by an operator.An operator

is a mathematical instruction: “Do something to this function or these num-

bers.” In other words, an operator acts on a function (or functions) to produce

a function. (Constants are special types of functions, ones that do not change

value.)

For example, in the equation 2 3 6, the operation is multiplication and

the operator is. It implies, “Multiply the two numbers together.” In fancier

terms, we can define the multiplication operation with some symbol, desig-

nated Mˆ(a,b). Its definition can be “Take two numbers and multiply them to-

gether.” Therefore,

Mˆ(2, 3)  6

is our fancy way of writing multiplication.Mˆis our multiplication operator,

where the ^ signifies an operator.

Operators can operate on functions as well as numbers. Consider the dif-

ferentiation of a simple function,F(x)  3 x^3  4 x^2 5, with respect to x:

d

d

x

(3x^3  4 x^2 5)  9 x^2  8 x

This could also be represented using simply F(x) to represent the function:

d

d

x

F(x)  9 x^2  8 x

The operator is d/dx, and can be represented by some symbol, say Dˆ, so that

the above expression can be simplified to

ˆD[F(x)]  9 x^2  8 x

The operator operated on a function and generated another function. It

is common to use a symbol to represent an operator, because some opera-

tors can have relatively complex forms. In applying a more complicated

mathematical operation, say (h^2 /8 

2 m)(d^2 /dx^2 ), to a wavefunction ,we

could write

8



h

(^2) m
2
d
d
x
2
2 
or, by defining the operator (h^2 /8
2 m)(d^2 /dx^2 ) as ˆT, we can rewrite the
above as simply
ˆT
which is much more compact. The above expression simply means “Take the
group of mathematical operations indicated by (h^2 /8
2 m)(d^2 /dx^2 ) and per-
form them on the wavefunction indicated by .” Performance of an operation
typically yields some expression, either a number or a function.
276 CHAPTER 10 Introduction to Quantum Mechanics