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## Schrödinger Equation (Time-Dependent Form)

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the wavefunction $\Psi(\mathbf{r}, t)$ of a particle changes over time. The time-dependent Schrödinger equation is given by:

### 1. Basic Form

$$ i\hbar \frac{\partial \Psi(\mathbf{r}, t)}{\partial t} = \hat{H} \Psi(\mathbf{r}, t) $$

where,
- $i$ is the imaginary unit,
- $\hbar$ is the reduced Planck constant ($\hbar = \frac{h}{2\pi}$),
- $\Psi(\mathbf{r}, t)$ is the wavefunction as a function of time,
- $\hat{H}$ is the Hamiltonian operator representing the total energy of the system.

### 2. Role of the Hamiltonian

The Hamiltonian $\hat{H}$ represents the total energy of the system, typically expressed as the sum of kinetic and potential energies. The Hamiltonian can be defined as follows:

$$ \hat{H} = \hat{T} + \hat{V} $$

where $\hat{T}$ is the kinetic energy operator and $\hat{V}$ is the potential energy operator.

### 3. Meaning of the Schrödinger Equation

The Schrödinger equation is an essential tool for understanding the dynamics of quantum systems. Through this equation, we can predict the state and time evolution of quantum systems. In particular, it allows us to determine how a system will evolve under given initial conditions.

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