Introduction to Schrödinger Equation
The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time.

Basic Formulation
- The equation can be expressed in two forms: the time-dependent and time-independent Schrödinger equations.
- The time-dependent form is given as:
\[
i\hbar \frac{\partial \Psi(\mathbf{r}, t)}{\partial t} = \hat{H} \Psi(\mathbf{r}, t)
\]
- Here, \( \Psi(\mathbf{r}, t) \) is the wave function, \( \hat{H} \) is the Hamiltonian operator, and \( \hbar \) is the reduced Planck's constant.

Derivation Steps
1. **Wave Function Concept**:
- In quantum mechanics, particles are described by wave functions that provide information about the probability amplitude of a system.
2. **Energy Representation**:
- The Hamiltonian operator \( \hat{H} \) represents the total energy (kinetic + potential) of the system.
3. **Postulate of Quantum Mechanics**:
- The change in wave function over time can be derived from the energy operator acting on the wave function.
4. **Time Evolution**:
- The time evolution is governed by the principle that the wave function must satisfy the linearity and superposition principles.
5. **Substituting the Operators**:
- By substituting the expressions for kinetic and potential energy into the Hamiltonian, we arrive at the time-independent form:
\[
\hat{H} \Psi = E \Psi
\]
- Where \( E \) is the energy eigenvalue.

Conclusion
The Schrödinger equation encapsulates the core principles of quantum mechanics, allowing the analysis of various quantum systems and phenomena. Understanding its derivation is crucial for deeper insights into quantum behavior.