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
Consider $\varphi$(x,t) is an unnormalized wave function. We can construct a normalized wave function as $\psi\left(x,t\right)$=N$\varphi$(x,t) where, N is the normalization constant. Therefore, $\int_{-\infty}^{\infty}\psi^\ast\left(x,t\right)\psi\left(x,t\right)$ = $|N^{2}| \intop_{-\infty}^{\infty}\varphi^{*}\left(x,t\right)\varphi\left(x,t\right)dx$=1 ⟹$N=\frac{1}{\sqrt{\intop_{-\infty}^{\infty}\varphi^{*}\left(x,t\right)\varphi\left(x,t\right)dx}}$ Example: Normalize the wave function given by $\psi\left(x\right)=Ne^{-\alpha\chi}\left(x>0\right)$ =$Ne^{\alpha\chi}\left(x<0\right)$ Solution. Normalization condition is $\intop_{-\infty}^{\infty}\psi^{*}\psi dx$=1 ⟹$|N^{2}| \left[\intop_{-\infty}^{0}e^{2\alpha\chi}dx+\intop_{0}^{\infty}e^{-2\alpha\chi}dx\right]=1$ ⟹$|N^{2}|\left[\frac{1}{2\alpha}+\frac{1}{2\alpha}\right]=1⟹N=\sqrt{\alpha}$
