Generalized Schrödinger Representation

In the preceding section, we developed the Schrödinger representation for the case of a single operator $x$ corresponding to a classical Cartesian coordinate. However, this scheme can easily be extended. Consider a system with $N$ generalized coordinates, $q_1\cdots q_N$ , which can all be simultaneously measured. These are represented as $N$ commuting operators, $q_1\cdots q_N$ , each with a continuous range of eigenvalues, $q_1'\cdots q_N'$ . Ket space is conveniently spanned by the simultaneous eigenkets of $q_1\cdots q_N$ , which are denoted $\vert q_1'\cdots q_N'\rangle$ . These eigenkets must form a complete set, otherwise the $q_1\cdots q_N$ would not be simultaneously observable.

The orthogonality condition for the eigenkets [i.e., the generalization of Equation (117)] is

$$\langle q_1'\cdots q_N'\vert q_1''\cdots q_N''\rangle = \delta(q_1'-q_1'')\,\delta(q_2'-q_2'')\cdots \delta(q_N'-q_N'').$$ (150)

The completeness condition [i.e., the generalization of Equation (118)] is

$$\int_{-\infty}^{+\infty} \cdots\int_{-\infty}^{+\infty} dq_1' \cdots dq_N'\, \vert q_1'\cdots q_N'\rangle \langle q_1'\cdots q_N'\vert = 1.$$ (151)

The standard ket $\rangle$ is defined such that

$$\langle q_1'\cdots q_N'\vert\rangle = 1.$$ (152)

The standard bra $\langle$ is the dual of the standard ket. A general state ket is written

$$\psi(q_1\cdots q_N)\rangle.$$ (153)

The associated wavefunction is

$$\psi(q_1'\cdots q_N') = \langle q_1'\cdots q_N'\vert\psi\rangle.$$ (154)

Likewise, a general state bra is written

$$\langle \phi(q_1\cdots q_N),$$ (155)

where

$$\phi(q_1'\cdots q_N') = \langle \phi\vert q_1'\cdots q_N'\rangle.$$ (156)

The probability of an observation of the system simultaneously finding the first coordinate in the range $q_1'$ to $q_1'+dq_1'$ , the second coordinate in the range $q_2'$ to $q_2' +dq_2'$ , etc., is

$$P(q_1'\cdots q_N'; dq_1'\cdots dq_N') = \vert\psi(q_1'\cdots q_N')\vert^{\,2}\,dq_1'\cdots dq_N'.$$ (157)

Finally, the normalization condition for a physical wavefunction is

$$\int_{-\infty}^{+\infty} \cdots \int_{-\infty}^{+\infty}dq_1'\cdots dq_N' \, \vert\psi(q_1'\cdots q_N')\vert^{\,2}= 1.$$ (158)

The $N$ linear operators $\partial/\partial q_i$ (where $i$ runs from 1 to $N$ ) are defined

$$\frac{\partial}{\partial q_i} \,\psi\rangle =\frac{\partial\psi}{\partial q_i} \rangle.$$ (159)

These linear operators can also act on bras (provided the associated wavefunctions are square integrable) in accordance with [see Equation (133)]

$$\langle \phi\, \frac{\partial }{\partial q_i} = -\langle \frac{\partial \phi} {\partial q_i}.$$ (160)

Corresponding to Equation (137), we can derive the commutation relations

$$\frac{\partial}{\partial q_i}\, q_j - q_j \,\frac{\partial}{\partial q_i} = \delta_{ij}.$$ (161)

It is also clear that

$$\frac{\partial}{\partial q_i}\,\frac{\partial}{\partial q_j}\, \p... ... \frac{\partial }{\partial q_j} \,\frac{\partial }{\partial q_i}\, \psi\rangle,$$ (162)

showing that

$$\frac{\partial}{\partial q_i}\,\frac{\partial}{\partial q_j} = \frac{\partial }{\partial q_j}\, \frac{\partial }{\partial q_i}.$$ (163)

It can be seen, by comparison with Equations (114)-(116), that the linear operators $-{\rm i}\,\hbar\, \partial/\partial q_i$ satisfy the same commutation relations with the $q$ 's and with each other that the $p$ 's do. The most general conclusion we can draw from this coincidence of commutation relations is (see Dirac)

$$p_i = -{\rm i} \,\hbar \frac{\partial}{\partial q_i} + \frac{\partial F(q_1\cdots q_N) }{\partial q_i}.$$ (164)

However, the function $F$ can be transformed away via a suitable readjustment of the phases of the basis eigenkets (see Section 2.4, and Dirac). Thus, we can always construct a set of simultaneous eigenkets of $q_1\cdots q_N$ for which

$$p_i = -{\rm i}\,\hbar \frac{\partial}{\partial q_i}.$$ (165)

This is the generalized Schrödinger representation.

It follows from Equations (152), (159), and (165) that

$$p_i \rangle = 0.$$ (166)

Thus, the standard ket in the Schrödinger representation is a simultaneous eigenket of all the momentum operators belonging to the eigenvalue zero. Note that

$$\langle q_1'\cdots q_N'\vert\,\frac{\partial}{\partial q_i}\, \ps... ...'} = \frac{\partial } {\partial q_i'}\,\langle q_1'\cdots q_N'\vert\psi\rangle.$$ (167)

Hence,

$$\langle q_1'\cdots q_N'\vert\, \frac{\partial}{\partial q_i} = \frac{\partial}{\partial q_i'}\, \langle q_1'\cdots q_N'\vert,$$ (168)

so that

$$\langle q_1'\cdots q_N'\vert\, p_i = -{\rm i}\,\hbar\,\frac{\partial}{\partial q_i'}\, \langle q_1'\cdots q_N'\vert.$$ (169)

The dual of the above equation gives

$$p_i\,\vert q_1'\cdots q_N'\rangle = {\rm i}\,\hbar\,\frac{\partial}{\partial q_i'}\, \vert q_1'\cdots q_N'\rangle.$$ (170)