# 2.3: Wavefunctions

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

(123) |

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

(124) |

giving

(125) |

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

(126) |

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.

### Contributors

- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)