Consider a simple system with one classical degree of freedom, which corresponds to the Cartesian coordinate . Suppose that is free to take any value (e.g., could be the position of a free particle). The classical dynamical variable is represented in quantum mechanics as a linear Hermitian operator which is also called . Moreover, the operator possesses eigenvalues lying in the continuous range (since the eigenvalues correspond to all the possible results of a measurement of ). We can span ket space using the suitably normalized eigenkets of . An eigenket corresponding to the eigenvalue is denoted . Moreover, [see Equation (85)]
The eigenkets satisfy the extremely useful relation [see Equation (87)]
This formula expresses the fact that the eigenkets are complete, mutually orthogonal, and suitably normalized.
A state ket (which represents a general state of the system) can be expressed as a linear superposition of the eigenkets of the position operator using Equation (118). Thus,
The quantity is a complex function of the position eigenvalue . We can write
Here, is the famous wavefunction of quantum mechanics. Note that state is completely specified by its wavefunction [because the wavefunction can be used to reconstruct the state ket using Equation (119)]. It is clear that the wavefunction of state is simply the collection of the weights of the corresponding state ket , when it is expanded in terms of the eigenkets of the position operator. Recall, from Section 1.10, that the probability of a measurement of a dynamical variable yielding the result when the system is in state is given by , assuming that the eigenvalues of are discrete. This result is easily generalized to dynamical variables possessing continuous eigenvalues. In fact, the probability of a measurement of yielding a result lying in the range to when the system is in a state is . In other words, the probability of a measurement of position yielding a result in the range to when the wavefunction of the system is is
This formula is only valid if the state ket is properly normalized: i.e., if . The corresponding normalization for the wavefunction is
Consider a second state represented by a state ket and a wavefunction . The inner product can be written
where use has been made of Equations (118) and (120). Thus, the inner product of two states is related to the overlap integral of their wavefunctions.
Consider a general function of the observable [e.g., ]. If then it follows that
where use has been made of Equation (117). Here, is the same function of the position eigenvalue that is of the position operator : i.e., if then . It follows, from the above result, that a general state ket can be written
where is the same function of the operator that the wavefunction is of the position eigenvalue , and the ket has the wavefunction . The ket is termed the standard ket. The dual of the standard ket is termed the standard bra, and is denoted . It is easily seen that
Note, finally, that is often shortened to , leaving the dependence on the position operator tacitly understood.
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)