Bottom Science- 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
If an operator  operating on a function: $\psi_{n}(x)$ gives $Â\psi_{n}(x) = Â\psi_{n}(x),$ then $\psi_{n}(x)$ is called the eigenfunction of  corresponding to eigenvalue $\lambda$ and the above equation is known as the eigenvalue equation of operator Â. Example: $ = \frac{d^{2}}{dx^{2}}$ and $\psi_{n}(x) = \alpha e^{-2x}$ $\Longrightarrow Â\psi_{n}(x)= \frac{d^{2}}{dx^{2}} (\alpha e^{-2x}) = 4\alpha e^{-2x} = 4\psi_{n}(x)$ $\psi_{n}(x)$ is an eigenfunction of  corresponding to eigenvalue 4. If the commutator bracket of two operators  and $\hat{B}$ is equal to zero i.e. $[Â,\hat{B}] = 0,$ then the physical observables corresponding to these operators are simultaneously accurately measurable and they have a complete set of simultaneous eigenfunctions.
