Displacement Operators

The situation is not so clear with state kets. The final state
of the system only determines the *direction* of the displaced state ket. Even if
we adopt the convention that all state kets have unit norms, the final ket is
still not completely determined, because it can be multiplied by a constant phase-factor. However, we know that the superposition relations between states
remain invariant under the displacement. This follows because the superposition
relations have a physical significance that is unaffected by a displacement of
the system.
Thus, if

in the undisplaced system, and the displacement causes ket to transform to ket , etc., then in the displaced system we have

Incidentally, this determines the displaced kets to within a single arbitrary phase-factor to be multiplied into all of them. The displaced kets cannot be multiplied by individual phase-factors, because this would wreck the superposition relations.

Since Equation (195) holds in the displaced system whenever Equation (194) holds in the undisplaced system, it follows that the displaced ket must be the result of some linear operator acting on the undisplaced ket . In other words,

(196) |

where an operator that depends only on the nature of the displacement. The arbitrary phase-factor by which all displaced kets may be multiplied results in being undetermined to an arbitrary multiplicative constant of modulus unity.

We now adopt the ansatz that any combination of bras, kets, and dynamical variables that possesses a physical significance is invariant under a displacement of the system. The normalization condition

(197) |

for a state ket certainly has a physical significance. Thus, we must have

(198) |

Now, and , so

(199) |

Because this must hold for any state ket , it follows that

Hence, the displacement operator is

(201) |

The equation

(202) |

where the operator represents a dynamical variable, has some physical significance. Thus, we require that

(203) |

where is the displaced operator. It follows that

(204) |

Since this is true for any ket , we have

Note that the arbitrary numerical factor in does not affect either of the results (200) and (205).

Suppose, now, that the system is displaced an *infinitesimal* distance
along the
-axis. We expect that the displaced ket
should
approach the undisplaced ket
in the limit as
. Thus,
we expect the limit

(206) |

to exist. Let

(207) |

where is denoted the

(208) |

where is the limit of . We have assumed, as seems reasonable, that tends to zero as . It is clear that the displacement operator is undetermined to an arbitrary imaginary additive constant.

For small , we have

It follows from Equation (200) that

(210) |

Neglecting order , we obtain

(211) |

Thus, the displacement operator is

(212) |

which implies

Let us consider a specific example. Suppose that a state has a wavefunction . If the system is displaced a distance along the -axis then the new wavefunction is (i.e., the same shape shifted in the -direction by a distance ). Actually, the new wavefunction can be multiplied by an arbitrary number of modulus unity. It can be seen that the new wavefunction is obtained from the old wavefunction according to the prescription . Thus,

(214) |

A comparison with Equation (213), using , yields

(215) |

It follows that obeys the same commutation relation with that , the momentum conjugate to , does [see Equation (116)]. The most general conclusion we can draw from this observation is that

(216) |

where is Hermitian (since is Hermitian). However, the fact that is undetermined to an arbitrary additive imaginary constant (which could be a function of ) enables us to transform the function out of the above equation, leaving

Thus, the displacement operator in the -direction is proportional to the momentum conjugate to . We say that is the

A finite translation along the -axis can be constructed from a series of very many infinitesimal translations. Thus, the operator which translates the system a distance along the -axis is written

(218) |

where use has been made of Equations (209) and (217). It follows that

The unitary nature of the operator is now clearly apparent.

We can also construct displacement operators which translate the system along
the
- and
-axes. Note that a displacement a distance
along the
-axis *commutes* with a displacement a distance
along the
-axis.
In other words, if the system is moved
along the
-axis, and then
along the
-axis, then it ends up in the same state as if it were moved
along the
-axis, and then
along the
-axis. The fact that
translations in independent directions commute is clearly associated with the
fact that the conjugate momentum operators
associated with these directions also commute
[see Equations (115) and (219)].