- Origin of Quantum Physics
- Wave function
- Collapse of Wave Function
- Physically Accepted Wave function
- Normalization Explained
- Method of Normalization
- Orthogonality & Orthonormality
- Hilbert Space
- Quantization Rules
- Operator Formalism
- Commutator Bracket
- Linear Operator
- Hermitian Operator
- Projection Operator
- Unitary Operator
- Parity Operator
- Expectation Value
- Schrodinger Equation
- Wave-Particle Duality Using Schrodinger Equation
- Superposition of States
- Various Representations of Wave Function
- Probability Current Density
- Uncertainty in Operators
- Shortcut for Calculating Momentum Expectation Value
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. 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. 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.Mathematical Explanation
Adjoint Operator
How to Identify a Linear Operator