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Operators
An operator, (say), is a mathematical entity which transforms
one function into another: i.e.,

(184) 
For instance, is an operator, since is a different function
to , and is fully specified once is given. Furthermore,
is also an operator, since is a different function
to , and is fully specified once is given.
Now,

(185) 
This can also be written

(186) 
where the operators are assumed to act on everything to
their right, and a final is understood [where is a general function]. The above expression illustrates
an important point: i.e., in general, operators do not
commute. Of course, some operators do commute: e.g.,

(187) 
Finally, an operator, , is termed linear if

(188) 
where is a general function, and a general complex number.
All of the operators employed in quantum mechanics are linear.
Now, from Eqs. (158) and (174),
These expressions suggest a number of things. First, classical dynamical
variables, such as and , are represented in quantum mechanics
by linear operators which act on the wavefunction. Second,
displacement is represented by the algebraic operator ,
and momentum by the differential operator
: i.e.,

(191) 
Finally, the expectation value of some dynamical variable represented by
the operator is simply

(192) 
Clearly, if an operator is to represent a dynamical variable which has
physical significance then its expectation value must be real.
In other words, if the operator represents a physical variable
then we require that
, or

(193) 
where is the complex conjugate of . An operator which
satisfies the above constraint is called an Hermitian operator.
It is easily demonstrated that and are both Hermitian.
The Hermitian conjugate, , of
a general operator, , is defined as follows:

(194) 
The Hermitian conjugate of an Hermitian operator is the same as the operator
itself: i.e., . For a nonHermitian operator, (say),
it is easily demonstrated that
, and that the operator is Hermitian.
Finally, if and are two operators, then
.
Suppose that we wish to find the operator which corresponds to the
classical dynamical variable . In classical mechanics, there
is no difference between and . However, in quantum
mechanics, we have already seen that . So,
should be choose or ? Actually, neither of these combinations
is Hermitian. However,
is Hermitian.
Moreover,
, which neatly resolves
our problem of which order to put and .
It is a reasonable guess that the operator corresponding to energy (which is
called the Hamiltonian, and conventionally denoted ) takes the form

(195) 
Note that is Hermitian. Now, it follows from Eq. (191) that

(196) 
However, according to Schrödinger's equation, (137), we have

(197) 
so

(198) 
Thus, the timedependent Schrödinger equation can be written

(199) 
Finally, if is a classical dynamical variable which is
a function of displacement, momentum, and energy, then a reasonable
guess for the corresponding operator in quantum mechanics is
, where
, and
.
Next: Momentum Representation
Up: Fundamentals of Quantum Mechanics
Previous: Ehrenfest's Theorem
Richard Fitzpatrick
20100720