The _[shift operator](Shift%20operator.md)_ or _translation operator_ in [quantum mechanics](Quantum%20Mechanics%20(index).md) models translations in space for a given [wavefunction,](Wavefunction.md) $\psi(\mathbf{x})$ where is $\mathbf{x}$ are spatial coordinate(s). ^5ad888 Like all operators that describe a [change](Unitary%20transformations%20in%20quantum%20mechanics.md) in wave function, the shift operator is [unitary](Unitary%20transformations%20in%20quantum%20mechanics.md#Unitary%20operators%20in%20quantum%20mechanics) and as such it is expressed as [a matrix exponential.](Unitary%20transformations%20in%20quantum%20mechanics.md#^7f7c4e) We define an operator that displaces a wavefunction by $\mathbf{r}$ as $\hat{U}_d(\mathbf{r})=e^{i\hat{p}\mathbf{r}}$ where $\hat{p}$ is the [momentum operator.](Momentum%20Operator.md) ^9f6787 Here, $\hat{p}$ may be thought of as the [generator](Momentum%20Operator.md#The%20momentum%20operator%20as%20the%20generator%20for%20the%20translation%20operator) for the translation operator. The form of the translation operator also follows from the general form of the [shift operator on Hilbert Spaces,](Shift%20operator.md#On%20Hilbert%20Spaces)  --- # Recommended reading The translation operator is succinctly introduced in the following exercise sheet: * [Mukhanov V., Homework 2, Quantum Mechanics 2 (2019/2020)](QM2_WS1920_ps2.pdf) Exercise 10. Here the the form of the displacement operator is shown by Taylor expanding it ([solution](sol2.pdf)). #QuantumMechanics/MathematicalFoundations #QuantumMechanics/QuantumDynamics