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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),

 (189) (190)

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 non-Hermitian 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 time-dependent 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 2010-07-20