Schrödinger equation from scratch
The wave-particle model states that matter and electromagnetic waves posess both particle and wave properties. The energy of a particle is quantized and is given by \(E=h\omega\), where \(h\) is the Planck's constant and \(\omega\) is the frequency of the electromagnetic wave associated with it. The momentum of the particle is given by \(\mathbf{p}=h\mathbf{k}\), where \(\mathbf{k}\) is the wavevector.
In quantum mehcanics wave packets are used to describe free particles. The wave packets can be decomposed into a Fourier integral of plane waves of the form
\[\begin{equation} \psi(\mathbf{r}, t)\propto e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}. \label{eq:1} \end{equation}\]By setting \(h=1\) and using relations \(E=h\omega=\omega\) and \(\mathbf{p}=h\mathbf{k}=\mathbf{k}\) we get
\[\begin{equation} \psi(\mathbf{r}, t)=Ne^{i(\mathbf{p}\cdot\mathbf{r}-Et)}, \label{eq:2} \end{equation}\]where \(N\) is a normalization constant.
To extract the energy and momentum from the wave function we indentify the energy operator \(\hat{E}=i\dfrac{\partial}{\partial t}\) and the momentum operator \(\hat{p}=-i\nabla\) giving
\[\begin{equation*} \begin{split} \hat{E}\psi&=E\psi \\ \hat{p}\psi&=p\psi. \end{split} \end{equation*}\]The energy of a non-relativistic free particle in the classical description is described by Hamiltonian
\[\begin{equation} H=E=T+V=\frac{\mathbf{p}^2}{2m}+V, \label{eq:3} \end{equation}\]where \(K\) is the kinetic energy and \(V\) is the potential energy. The energy of a non-relativistic free particle in the quantum mechanical approach is obtained by substituting the momentum and energy in equation \ref{eq:3} with the operators defined earlier. We end up with
\[\begin{equation} i\dfrac{\partial\psi(\mathbf{r}, t)}{\partial t}=\hat{H}\psi(\mathbf{r}, t)=\left(-\frac{1}{2m}\nabla^2+\hat{V}\right)\psi(\mathbf{r}, t). \end{equation}\]