Schrödinger Representation

Any linear operator that acts on ket vectors can also act on bra vectors. Consider acting on a general bra . According to Equation (1.37), the bra satisfies

(2.38) |

Making use of Equations (2.27), (2.29), and (2.37) we can write

(2.39) |

The right-hand side can be transformed via integration by parts to give

assuming that the contributions from the limits of integration vanish. It follows that

(2.41) |

which implies that

The neglect of contributions from the limits of integration in Equation (2.40) is reasonable because physical wavefunctions are square-integrable. [See Equation (2.31).] Note that

(2.43) |

where use has been made of Equation (2.42). It follows, by comparison with Equations (1.38) and (2.36), that

Thus, is an anti-Hermitian operator.

Let us evaluate the commutation relation between the operators and . We have

(2.45) |

Because this holds for any ket , it follows that

Let be the momentum conjugate to (for the simple system under consideration, is a straightforward linear momentum). According to Equation (2.25), and satisfy the commutation relation

It can be seen, by comparison with Equation (2.46), that the Hermitian operator satisfies the same commutation relation with that does. The most general conclusion which may be drawn from a comparison of Equations (2.46) and (2.47) is that

because (as is easily demonstrated) a general function of the position operator automatically commutes with . (See Exercise 3.)

We have chosen to normalize the eigenkets and eigenbras of the position operator such that they satisfy the normalization condition (2.26). However, this choice of normalization does not uniquely determine the eigenkets and eigenbras. Suppose that we transform to a new set of eigenbras which are related to the old set via

where is a real function of . This transformation amounts to a rearrangement of the relative phases of the eigenbras. The new normalization condition is

(2.50) |

Thus, the new eigenbras satisfy the same normalization condition as the old eigenbras.

By definition, the standard ket satisfies . It follows from Equation (2.49) that the new standard ket is related to the old standard ket via

where is a real function of the position operator . The dual of the previous equation yields the transformation rule for the standard bra,

The transformation rule for a general operator follows from Equations (2.51) and (2.52), plus the requirement that the triple product remain invariant (which must be the case, otherwise the probability of a measurement yielding a certain result would depend on the choice of eigenbras). Thus,

Of course, if commutes with then is invariant under the transformation. In fact, is the only operator (that we know of) that does not commute with , so Equation (2.53) yields

(2.54) |

where the subscript ``old'' is taken as read. It follows, from Equation (2.48), that the momentum operator can be written

(2.55) |

Thus, the special choice

yields

Equation (2.56) fixes to within an arbitrary additive constant. In other words, the special eigenkets and eigenbras for which Equation (2.57) is true are determined to within an arbitrary common phase-factor.

In conclusion, it is possible to find a set of basis eigenkets and eigenbras of the position operator that satisfy the normalization condition (2.26), and for which the momentum conjugate to can be represented as the operator

(2.58) |

A general state ket is written , where the standard ket satisfies , and where is the wavefunction. This scheme is known as the