Translation operator (quantum mechanics)
It is a special case of the shift operator from functional analysis.
Because of this relationship, conservation of momentum holds when the translation operators commute with the Hamiltonian, i.e. when laws of physics are translation-invariant.
An alternative (and equivalent) way to describe what the translation operator determines is based on position-space wavefunctions.
In introductory physics, momentum is usually defined as mass times velocity.
This is more specifically called canonical momentum, and it is usually but not always equal to mass times velocity.
One notable exception pertains to a charged particle in a magnetic field in which the canonical momentum includes both the usual momentum and a second terms proportional to the magnetic vector potential.
We suppose that for an infinitesimal translation that the higher-order terms in the series become successively smaller.
With this preliminary result, we proceed to write the an infinite amount of infinitesimal actions as
is the operator exponential and the right-hand side is the Taylor series expansion.
is an infinitesimal transformation, generating translations of the real line via the exponential.
It is for this reason that the momentum operator is referred to as the generator of translation.
[2] A nice way to double-check that these relations are correct is to do a Taylor expansion of the translation operator acting on a position-space wavefunction.
Expanding the exponential to all orders, the translation operator generates exactly the full Taylor expansion of a test function:
, i.e. a translation by a distance of 0 is the same as the identity operator which leaves all states unchanged.
Detailed proofs of this can be found in many textbooks and online (e.g. https://physics.stackexchange.com/a/832341/194354).
, with the operation of multiplication defined as the result of successive translations (i.e. function composition), satisfies all the axioms of a group: Therefore, the set
In the quantum formulation, the expectation values[5] play the role of the classical variables.
This result is consistent with what you would expect from an operation that shifts the particle by that amount.
On the other hand, when the translation operator acts on a state, the expectation value of the momentum is not changed.
In quantum mechanics, the Hamiltonian is the operator corresponding to the total energy of a system.
be the continuous translation operator that shifts all particles and fields in the universe simultaneously by the same amount.
If we assert the a priori axiom that this translation is a continuous symmetry of the Hamiltonian (i.e., that
is independent of location), then, as a consequence, conservation of momentum is universally valid.
In summary, whenever the Hamiltonian for a system remains invariant under continuous translation, then the system has conservation of momentum, meaning that the expectation value of the momentum operator remains constant.
In general, the Hamiltonian is not invariant under any translation represented by
Therefore, the Hamiltonian is invariant under such translation (which no longer remains continuous).
The ions in a perfect crystal are arranged in a regular periodic array.
Moreover, the ions are not in fact stationary, but continually undergo thermal vibrations about their equilibrium positions.
To deal with this type of problems the main problem is artificially divided in two parts: (a) the ideal fictitious perfect crystal, in which the potential is genuinely periodic, and (b) the effects on the properties of a hypothetical perfect crystal of all deviations from perfect periodicity, treated as small perturbations.
It can be concluded that the translational invariance of Hamiltonian implies that the same experiment repeated at two different places will give the same result (as seen by the local observers).
