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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

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

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

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

The standard ket $ \rangle$ is defined such that

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

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

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

The associated wavefunction is

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

Likewise, a general state bra is written

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


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

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

Finally, the normalization condition for a physical wavefunction is

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

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

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

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

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

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

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

It is also clear that

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

showing that

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

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

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

This is the generalized Schrödinger representation.

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

$\displaystyle p_i \rangle = 0.$ (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

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


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

so that

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

The dual of the above equation gives

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

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Next: Momentum Representation Up: Position and Momentum Previous: Schrödinger Representation
Richard Fitzpatrick 2013-04-08