In quantum mechanics, a translation operator is defined as an operator which shifts particles and fields by a certain amount in a certain direction. It is a special case of the shift operator from functional analysis.
More specifically, for any displacement vector , there is a corresponding translation operator that shifts particles and fields by the amount .
For example, if acts on a particle located at position , the result is a particle at position .
Translation operators are unitary.
Translation operators are closely related to the momentum operator; for example, a translation operator that moves by an infinitesimal amount in the direction has a simple relationship to the -component of the momentum operator. 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. This is an example of Noether's theorem.
Action on position eigenkets and wavefunctions
The translation operator moves particles and fields by the amount . Therefore, if a particle is in an eigenstate of the position operator (i.e., precisely located at the position ), then after acts on it, the particle is at the position :
An alternative (and equivalent) way to describe what the translation operator determines is based on position-space wavefunctions. If a particle has a position-space wavefunction , and acts on the particle, the new position-space wavefunction is defined by
This relation is easier to remember as which can be read as: "The value of the new wavefunction at the new point equals the value of the old wavefunction at the old point".Lecture notes by Robert Littlejohn
Here is an example showing that these two descriptions are equivalent. The state corresponds to the wavefunction (where is the Dirac delta function), while the state corresponds to the wavefunction These indeed satisfy
Momentum as generator of translations
In introductory physics, momentum is usually defined as mass times velocity. However, there is a more fundamental way to define momentum, in terms of translation operators. 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 term proportional to the magnetic vector potential. This definition of momentum is especially important because the law of conservation of momentum applies only to canonical momentum, and is not universally valid if momentum is defined instead as mass times velocity (the so-called "kinetic momentum"), for reasons explained below.
The (canonical) momentum operator is defined as the gradient of the translation operators near the origin:
where is the reduced Planck constant. For example, what is the result when the operator acts on a quantum state? To find the answer, translate the state by an infinitesimal amount in the -direction, calculate the rate that the state is changing, and multiply the result by . For example, if a state does not change at all when it is translated an infinitesimal amount the -direction, then its -component of momentum is 0.
More explicitly, is a vector operator (i.e. a vector operator consisting of three operators ), components is given by:
where is the identity operator and is the unit vector in the -direction. ( and are defined analogously.)
The equation above is the most general definition of . In the special case of a single particle with wavefunction , can be written in a more specific and useful form. In one dimension:
While in three dimensions,
as an operator acting on position-space wavefunctions. This is the familiar quantum-mechanical expression for , but we have derived it here from a more basic starting point.
We have now defined in terms of translation operators. It is also possible to write a translation operator as a function of . The method consists of considering an infinitesimal action on a wavefunction, and expanding the transformed wavefunction as a sum of the initial wavefunction and a first-order perturbative correction; and then expressing a finite translation as a huge number of consecutive tiny translations, and then use the fact that infinitesimal translations can be written in terms of . From what has been stated previously, we know from above that if acts on that the result is
The right-hand side may be written as a Taylor series
We suppose that for an infinitesimal translation that the higher-order terms in the series become successively smaller. From which we write
With this preliminary result, we proceed to write the an infinite amount of infinitesimal actions as
The right-hand side is precisely a series for an exponential. Hence,
where is the operator exponential and the right-hand side is the Taylor series expansion. For very small , one can use the approximation:
The operator equation
is an operator version of Taylor's theorem; and is, therefore, only valid under caveats about being an analytic function. Concentrating on the operator part, it shows that 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.
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:
So every translation operator generates exactly the expected translation on a test function if the function is analytic in some domain of the complex plane.
Properties
Successive translations
In other words, if particles and fields are moved by the amount and then by the amount , overall they have been moved by the amount . For a mathematical proof, one can look at what these operators do to a particle in a position eigenstate:
Since the operators and have the same effect on every state in an eigenbasis, it follows that the operators are equal.
Identity translation
The translation , i.e. a translation by a distance of 0 is the same as the identity operator which leaves all states unchanged.
Inverse
The translation operators are invertible, and their inverses are:
This follows from the "successive translations" property above, and the identity translation.
Translation operators commute with each other
because both sides are equal to .
Translation operators are unitary
To show that translation operators are unitary, we first must prove that the momentum operator is Hermitian. Then, we can prove that the translation operator meets two criteria that are necessary to be a unitary operator.
To begin with, the linear momentum operator is the rule that assigns to any in the domain the one vector
in the codomain is. Since
therefore the linear momentum operator is, in fact, a Hermitian operator. Detailed proofs of this can be found in many textbooks and online (e.g. https://physics.stackexchange.com/a/832341/194354).
Having in hand that the momentum operator is Hermitian, we can prove that the translation operator is a unitary operator. First, it must shown that translation operator is a bounded operator. It is sufficient to state that for all that
Second, it must be (and can be) shown that
A detailed proof can be found in reference https://math.stackexchange.com/a/4990451/309209.
Translation Operator operating on a bra A translation operator operating on a bra in the position eigenbasis gives:
Splitting a translation into its components
According to the "successive translations" property above, a translation by the vector can be written as the product of translations in the component directions:
where are unit vectors.
Commutator with position operator
Suppose is an eigenvector of the position operator with eigenvalue . We have
while
Therefore, the commutator between a translation operator and the position operator is:
This can also be written (using the above properties) as:
where is the identity operator.
Commutator with momentum operator
Since translation operators all commute with each other (see above), and since each component of the momentum operator is a sum of two scaled translation operators (e.g. ), it follows that translation operators all commute with the momentum operator, i.e.
This commutation with the momentum operator holds true generally even if the system is not isolated where energy or momentum may not be conserved.
Translation group
The set of translation operators for all , with the operation of multiplication defined as the result of successive translations (i.e. function composition), satisfies all the axioms of a group: Closure When two translations are done consecutively, the result is a single different translation. (See "successive translations" property above.) Existence of identity A translation by the vector is the identity operator, i.e. the operator that has no effect on anything. It functions as the identity element of the group. Every element has an inverse As proven above, any translation operator is the inverse of the reverse translation . Associativity This is the claim that . It is true by definition, as is the case for any group based on function composition.
Therefore, the set of translation operators for all forms a group.Page-816, Chapter-17, Mathematical Methods for Physicists, Seventh Edition, by Arfken, Weber and Harris Since there are continuously infinite number of elements, the translation group is a continuous group. Moreover, the translation operators commute among themselves, i.e. the product of two translation (a translation followed by another) does not depend on their order. Therefore, the translation group is an abelian group.Page-47, Chapter-1, Modern Quantum Mechanics, Second edition, J.J. Sakurai, Jim J. Napolitano
The translation group acting on the Hilbert space of position eigenstates is isomorphic to the group of vector additions in the Euclidean space.
Expectation values of position and momentum in the translated state
Consider a single particle in one dimension. Unlike classical mechanics, in quantum mechanics a particle neither has a well-defined position nor a well-defined momentum. In the quantum formulation, the expectation valuesP. 127, Section 4.2, R. Shankar, Principles of Quantum Mechanics play the role of the classical variables. For example, if a particle is in a state , then the expectation value of the position is , where is the position operator.
If a translation operator acts on the state , creating a new state then the expectation value of position for is equal to the expectation value of position for plus the vector . 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. This can be proven in a similar way as the above, but using the fact that translation operators commute with the momentum operator. This result is again consistent with expectations: translating a particle does not change its velocity or mass, so its momentum should not change.
Translational invariance
In quantum mechanics, the Hamiltonian is the operator corresponding to the total energy of a system. For any in the domain, let the one vector in the codomain be a newly translated state. If
then a Hamiltonian is said to be invariant. Since the translation operator is a unitary operator, the antecedent can also be written as
Since this holds for any in the domain, the implication is that
or that