- 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

The wave-particle duality is a fundamental concept in quantum physics that states that subatomic particles, such as electrons and photons, can exhibit both wave-like and particle-like behavior.

The mathematical description of this duality is given by wave functions, which are complex mathematical functions that describe the probability of finding a particle in a certain location or with a certain momentum.

One of the most important mathematical descriptions of the wave-particle duality is the Schrödinger equation. The time-independent Schrödinger equation is given by:

$\large{-\frac{\hbar^2}{2m}\nabla^2\psi(x) + V(x)\psi(x) = E\psi(x)}$

where $\hbar$ is the reduced Planck constant, $m$ is the mass of the particle, $\nabla^2$ is the Laplacian operator, $V(x)$ is the potential energy of the particle, $E$ is the total energy of the particle and $\psi(x)$ is the wave function.

The solution to the Schrödinger equation gives the probability amplitude of finding a particle in a particular location.

The time-dependent Schrödinger equation is given by:

$\large{i\hbar\frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi + V(x)\psi}$

where $i$ is the imaginary unit. The solution to the time-dependent Schrödinger equation gives the probability amplitude of finding a particle in a particular location at a particular time.

The wave functions are complex-valued functions and the probability of finding a particle in a certain location is given by the modulus square of the wave function:

$\large{|\psi(x)|^2}$

This is known as the probability density function. The wave-particle duality can also be described by the Heisenberg uncertainty principle, which states that the position and momentum of a particle cannot be measured simultaneously with arbitrary precision. The uncertainty in position and momentum is given by:

$\large{\Delta x \Delta p \geq \frac{\hbar}{2}}$

The Schrödinger equation and the Heisenberg uncertainty principle are key mathematical tools that help to understand the wave-particle duality.