Consider a system with one degree of freedom, describable in terms of a coordinate and its conjugate momentum , both of which have a continuous range of eigenvalues. We have seen that it is possible to represent the system in terms of the eigenkets of . This is termed the Schrödinger representation. However, it is also possible to represent the system in terms of the eigenkets of .
Consider the eigenkets of which belong to the eigenvalues . These are denoted . The orthogonality relation for the momentum eigenkets is
and the corresponding completeness relation is
A general state ket can be written
where the standard ket satisfies
Note that the standard ket in this representation is quite different to that in the Schrödinger representation. The momentum space wavefunction satisfies
The probability that a measurement of the momentum yields a result lying in the range to is given by
Finally, the normalization condition for a physical momentum space wavefunction is
The fundamental commutation relations (114)-(116) exhibit a particular symmetry between coordinates and their conjugate momenta. If all the coordinates are transformed into their conjugate momenta, and vice versa, and is then replaced by , then the commutation relations are unchanged. It follows from this symmetry that we can always choose the eigenkets of in such a manner that the coordinate can be represented as (see Section 2.4)
This is termed the momentum representation.
The above result is easily generalized to a system with more than one degree of freedom. Suppose the system is specified by coordinates, , and conjugate momenta, . Then, in the momentum representation, the coordinates can be written as
We also have
The momentum representation is less useful than the Schrödinger representation for a very simple reason. The energy operator (i.e., the Hamiltonian) of most simple systems takes the form of a sum of quadratic terms in the momenta (i.e., the kinetic energy) plus a complicated function of the coordinates (i.e., the potential energy). In the Schrödinger representation, the eigenvalue problem for the energy translates into a second-order differential equation in the coordinates, with a complicated potential function. In the momentum representation, the problem transforms into a high-order differential equation in the momenta, with a quadratic potential. With the mathematical tools at our disposal, we are far better able to solve the former type of problem than the latter. Hence, the Schrödinger representation is generally more useful than the momentum representation.
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)