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
Orthogonality condition of wave functions: Two wave functions $\psi_{m}\left(x\right)$ and $\psi_{n}\left(x\right)$ are said to be orthogonal to each other, if $\intop_{-\infty}^{\infty}\psi_{m}^{*}(x)\psi_{n}(x)$dx=0 $(m\neq n)$ OR $<\psi_{m}|\psi_{n}> = 0 (m \neq n)$ i.e. if a particle is in the state $\psi_{m}(x)$ , then the particle cannot be in the state $\psi_{n}(x)$ simultaneously together. Orthonormality condition of wave functions: Two wave functions $\psi_{m}\left(x\right)$ and $\psi_{n}\left(x\right)$ are said to be orthonormal to each other, if they are orthogonal and also normalized. Orthonormal = Orthogonal + Normalized $\intop_{-\infty}^{\infty}\psi_{m}^{*}(x)\psi_{n}(x)$dx=0 $(m\neq n)$ $\intop_{-\infty}^{\infty}\psi_{m}^{*}(x)\psi_{n}(x)$dx=1 $(m= n)$ OR $<\psi_{m}|\psi_{n}> = 0 (m \neq n)$ $<\psi_{m}|\psi_{n}> = 1 (m = n)$
