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
As the time changes from t=0, the probability of finding the particle in some region of space may increase. If the probability increases in some region, then it should decrease in some other region such that the total probability of finding the particle in the entire space should be equal to one. We can assume this as a flow of probability from one region to another region, like a fluid or current. Therefore, the probability flow satisfies the equation of continuity i.e. $\overrightarrow{\nabla}.\overrightarrow{J}+\frac{\partial \rho}{\partial t}=0$ where $\rho$ = probability density $=\psi*\psi$ and $\overrightarrow{J}=$ probability current density $=-\frac{i\hbar}{2m}\left[\psi^{*}\overrightarrow{\nabla}\psi-\psi\overrightarrow{\nabla}\psi^{*}\right]$ The magnitude of probability current density represents the flux of the particle i.e. number of particles passing through per unit area per unit time and the direction of probability current density is along the direction of the flow of the particles.
