- 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, quantization is the process of restricting the possible values of a physical quantity to certain discrete values.

There are **two main rules for quantization** in quantum mechanics:

- the wave function quantization
- the energy quantization

**Wave Function Quantization**

The wave function quantization rule states that **the wave function of a particle must be normalized**, meaning that the integral of the squared modulus of the wave function over all space must be equal to 1.

Mathematically, this rule is expressed as:

$\large{\int |\psi(x)|^2 dx = 1}$

This rule ensures that the probability of finding a particle at a certain location is always between 0 and 1.

**Energy Quantization**

The energy quantization rule states that the **energy of a system can only take on certain discrete values**, and that any transition between energy levels must involve the absorption or emission of a quantum of energy.

This rule is mathematically expressed as:

$\large{E_n = nh\nu}$

where $E_n$ is the energy of the n-th level, $h$ is the Planck constant and $\nu$ is the frequency of the energy quantum.

This rule is derived from the time-dependent Schrödinger equation, which describes how the wave function of a system changes over time.

The solutions to the Schrödinger equation are called energy eigenstates, and they have the property that their energy remains constant over time.