# Brief Derivation of Schrödinger Equation

Imagine a particle constrained to move along the x-axis, subject to a potential energy $V(x,t)\!$. Classically, we could model this system by writing down its Hamiltonian $H,\!$ given by

$H = \frac{p^2}{2m} + V(x,t).$

We then employ Hamilton's equations of motion,

$\dot{p}=-\frac{\partial H}{\partial x},\,\dot{x}=\frac{\partial H}{\partial p},$

where a dot denotes a time derivative, to determine the motion of the particle. Applying these equations to the above Hamiltonian, we can recover Newton's second law,

$m\ddot{a}=-\frac{\partial V}{\partial x}=F.$

Now by applying the appropriate initial conditions for our particle, we obtain a solution for the trajectory of the particle. As we will see, the above relation is only an approximation to actual physical reality. As we attempt to describe increasingly smaller objects, we enter the quantum mechanical regime, where we can no longer neglect the particles' wave properties. Allowing $\displaystyle{p \rightarrow \frac{\hbar}{i}\frac{\partial}{\partial x}}$ and $\displaystyle{H \rightarrow i\hbar \frac{\partial}{\partial t}},$ we can use the Hamiltonian for a classical particle above to find an equation that describes this wave nature. We find that the wave function $\Psi(x,t)\!$ satisfies the Schrödinger equation for a scalar potential $V(x,t)\!$ in one dimension:

$i\hbar\frac{\partial}{\partial t}\Psi(x,t)=\left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x,t)\right]\Psi(x,t)$

A similar equation may be derived in multiple dimensions:

$i\hbar\frac{\partial}{\partial t}\Psi(\textbf{r},t)=\left[-\frac{\hbar^2}{2m}\nabla^2+V(\textbf{r},t)\right]\Psi(\textbf{r},t)$

Given a solution which satisfies the above Schrödinger equation, quantum mechanics provides a mathematical description of the laws obeyed by the probability amplitudes associated with quantum motion.

We can also generalize the Schrödinger equation to a system which contains $N \!$ particles. We assume that the wave function is $\Psi(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N, t)$ and that the Hamiltonian of the system can be expressed as

$H= \sum_{k=1}^N \frac{\textbf{p}^2_k}{2m_k}+V(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N).$

Making a similar replacement as before, we may obtain the Schrödinger equation for a many-particle system:

$i\hbar\frac{\partial}{\partial t}\Psi(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N, t)=\left[\sum_{k=1}^N \frac{\hbar^2}{2m_k}\nabla_{k}^2+V(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N)\right]\Psi(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N, t)$

Here, $\nabla_{k}^2$ is the Lapalacian operator that acts on the coordinates of particle $k.\!$