# Linear Operator | Quantum Mechanics

Quantum Mechanics
Basic Quantum Mechanics

Linear Operator

In quantum mechanics, a linear operator is a mathematical object that acts on a wave function to produce another wave function.

Linear operators are used to representing physical observables, such as position, momentum, and energy.

## Mathematical Explanation

A linear operator $\hat{A}$ can be written as:

$$\hat{A} \psi(x) = A(x) \psi(x)$$

where $\psi(x)$ is the wave function and $A(x)$ is the eigenvalue of the operator.

For example, the position operator $\hat{x}$ is a linear operator that acts on a wave function $\psi(x)$ to produce the position of the particle:

$$\hat{x} \psi(x) = x \psi(x)$$

Similarly, the momentum operator $\hat{p}$ is a linear operator that acts on a wave function $\psi(x)$ to produce the momentum of the particle:

$$\hat{p} \psi(x) = -i\hbar \frac{\partial}{\partial x} \psi(x)$$

where $\hbar$ is the reduced Planck constant.

In quantum mechanics, observables are represented by self-adjoint linear operators, which have the property that the adjoint operator is equal to the original operator.

The adjoint operator $\hat{A}^{\dagger}$ is defined as:

$$(\hat{A}^{\dagger} \psi)(x) = \psi^{*}(x) \hat{A} \psi(x)$$

where $\psi^{*}(x)$ is the complex conjugate of the wave function $\psi(x)$.

Self-adjoint operators have real eigenvalues and their eigenvectors form a complete orthonormal basis for the Hilbert space, which is the space of all possible wave functions.

## How to Identify a Linear Operator

An operator $Â$ is said to be a linear if

(i) $\hat{A}\left[c\psi(x)\right]=c\hat{A}\psi(x)$ and

(ii) $\hat{A}\left[\psi_{1}(x)+\psi_{2}(x)\right]=\hat{A}\psi_{1}(x)+\hat{A}\psi_{2}(x)$

All Quantum Mechanical Operators are linear in nature.