Wavefunctions

Consider a simple system with one classical degree of freedom, which corresponds to the Cartesian coordinate $x$ . Suppose that $x$ is free to take any value (e.g., $x$ could be the position of a free particle). The classical dynamical variable $x$ is represented in quantum mechanics as a linear Hermitian operator that is also called $x$ . Moreover, the operator $x$ possesses eigenvalues $x'$ lying in the continuous range $-\infty< x'<+\infty$ (because the eigenvalues correspond to all the possible results of a measurement of $x$ ). We can span ket space using the suitably normalized eigenkets of $x$ . An eigenket corresponding to the eigenvalue $x'$ is denoted $\vert x'\rangle$ . Moreover,

$$\langle x' \vert x''\rangle = \delta(x'-x'').$$ (2.26)

[See Equation (1.88).] The eigenkets satisfy the extremely useful relation

$$\int_{-\infty}^{+\infty} d x' \, \vert x'\rangle\langle x'\vert= 1.$$ (2.27)

[See Equation (1.94).] This formula expresses the fact that the eigenkets are complete, mutually orthogonal, and suitably normalized.

A state ket $\vert A\rangle$ (which represents a general state $A$ of the system) can be expressed as a linear superposition of the eigenkets of the position operator using Equation (2.27). Thus,

$$\vert A\rangle = \int_{-\infty}^{+\infty} dx' \,\langle x'\vert A\rangle \vert x'\rangle$$ (2.28)

The quantity $\langle x'\vert A\rangle$ is a complex function of the position eigenvalue $x'$ . We can write

$$\langle x'\vert A\rangle = \psi_A(x').$$ (2.29)

Here, $\psi_A(x')$ is the famous wavefunction of quantum mechanics [99,32]. Note that state $A$ is completely specified by its wavefunction $\psi_A(x')$ [because the wavefunction can be used to reconstruct the state ket $\vert A\rangle$ using Equation (2.28)]. It is clear that the wavefunction of state $A$ is simply the collection of the weights of the corresponding state ket $\vert A\rangle$ , 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 $\xi$ yielding the result $\xi'$ when the system is in (a properly normalized) state $A$ is given by $\vert\langle \xi'\vert A\rangle\vert^{\,2}$ , assuming that the eigenvalues of $\xi$ are discrete. This result is easily generalized to dynamical variables possessing continuous eigenvalues. In fact, the probability of a measurement of $x$ yielding a result lying in the range $x'$ to $x'+dx'$ when the system is in a state $\vert A\rangle$ is $\vert\langle x'\vert A\rangle\vert^{\,2}\,dx'$ . In other words, the probability of a measurement of position yielding a result in the range $x'$ to $x'+dx'$ when the wavefunction of the system is $\psi_A(x')$ is

$$P(x', dx') = \vert\psi_A(x')\vert^{\,2}\, dx'.$$ (2.30)

This formula is only valid if the state ket $\vert A\rangle$ is properly normalized: that is, if $\langle A\vert A\rangle =1$ . The corresponding normalization for the wavefunction is

$$\int_{-\infty}^{+\infty} dx'\, \vert\psi_A(x')\vert^{\,2}= 1.$$ (2.31)

Consider a second state $B$ represented by a state ket $\vert B\rangle$ and a wavefunction $\psi_B(x')$ . The inner product $\langle B\vert A \rangle$ can be written

$$\langle B\vert A\rangle = \int_{-\infty}^{+\infty} dx'\,\langle B... ...\vert A \rangle = \int_{-\infty}^{+\infty} dx'\,\psi_B^\ast (x') \,\psi_A'(x'),$$ (2.32)

where use has been made of Equations (2.27) and (2.29). Thus, the inner product of two states is related to the overlap integral of their wavefunctions.

Consider a general function $f(x)$ of the observable $x$ [e.g., $f(x)=x^{\,2}$ ]. If $\vert B\rangle = f(x)\,\vert A\rangle$ then it follows that

$$\psi_B(x') = \langle x'\vert f(x) \int_{-\infty}^{+\infty} dx''\,\psi_A(x'')\,\vert x''\rangle$$

$$= \int_{-\infty}^{+\infty} dx''\, f(x'')\,\psi_A(x'') \,\langle x'\vert x''\rangle,$$ (2.33)

giving

$$\psi_B(x') = f(x')\, \psi_A(x'),$$ (2.34)

where use has been made of Equation (2.26). (See Exercise 3.) Here, $f(x')$ is the same function of the position eigenvalue $x'$ that $f(x)$ is of the position operator $x$ . For instance, if $f(x)=x^{\,2}$ then $f(x') = x'^{\,2}$ . It follows, from the previous result, that a general state ket $\vert A\rangle$ can be written

$$\vert A\rangle = \psi_A(x) \rangle,$$ (2.35)

where $\psi_A(x)$ is the same function of the operator $x$ that the wavefunction $\psi_A(x')$ is of the position eigenvalue $x'$ , and the ket $\rangle$ has the wavefunction $\psi(x') =1$ . The ket $\rangle$ is termed the standard ket. The dual of the standard ket is termed the standard bra, and is denoted $\langle$ . It is easily seen that

$$\langle \psi_A^{\,\ast}(x) \stackrel{\rm DC}{\longleftrightarrow} \psi_A(x)\rangle.$$ (2.36)

Note, finally, that $\psi_A(x)\rangle$ is often shortened to $\psi_A\rangle$ , leaving the dependence on the position operator $x$ tacitly understood.