Consider a system with one degree of freedom corresponding to the Cartesian coordinate \(x\) . Suppose that we displace this system some distance along the \(x\) -axis. We could imagine that the system is on wheels, and we just give it a little push. The final state of the system is completely determined by its initial state, together with the direction and magnitude of the displacement. Note that the type of displacement we are considering is one in which everything to do with the system is displaced. So, if the system is subject to an external potential then the potential must be displaced.
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,
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
for a state ket certainly has a physical significance. Thus, we must have
Now, and , so
Because this must hold for any state ket , it follows that
Hence, the displacement operator is unitary. Note that the above relation implies that
where the operator represents a dynamical variable, has some physical significance. Thus, we require that
where is the displaced operator. It follows that
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
to exist. Let
where is denoted the displacement operator along the -axis. The fact that can be replaced by , where is a real phase-angle, implies that can be replaced by
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
Neglecting order , we obtain
Thus, the displacement operator is anti-Hermitian. Substituting into Equation (205), and again neglecting order , we find that
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,
A comparison with Equation (213), using , yields
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
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 generator of translations along the -axis.
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
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)].
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)