Erwin Schrödinger derived the Schrödinger equation that became fundamental part of quantum theory. Its solutions are wave functions.
It was discovered that light can behave as a wave and as a massless particle called a photon. Quantum theory is based on the principle that massive particles can also behave like waves.
A wave is described by its wavelength \(\lambda\) and its frequency \(\nu\). The energy \(E\) and momentum \(p\) of a particle can be defined using the Planck constant \(h \approx 6.6\cdot 10^{-34} Js\).
\[E = h\nu \quad p = h/\lambda\]
Waves can also alternatively be described by their wavenumber \(k = 2\pi/\lambda\) and their angular frequency \(\omega = 2\pi/\nu\). The energy and momentum can be defined using the reduced Planck constant \(\hbar = h/2\pi\).
\[E = \hbar\omega \quad p = \hbar k\]
The kinetic energy of a particle of ma ss \(m\) and velocity \(v\) is \(E = mv^2/2\). Using momentum in place of velocity gives \(E = p^2/2m\).
Given that the particle is moving along the \(x\) axis and has potential energy \(V(x, t)\), then the total energy is the sum of its kinetic enrgy and its potenial energy.
\[E = \frac{p^2}{2m} + V(x, t)\]
A wave function in one-dimension can be defined as \(\Psi(x, t) = A \cos(kx - \omega t + \phi)\). Where \(A\) is the wave amplitude and \(\phi\) is the wave phase. This can be written using complex numbers where \(C\) is a complex number containing both amplitude and phase.
\[\Psi(x, t) = Ce^{i(kx - \omega t)}\]
Substitute the wave energy and momentum equations into the total energy equation and multiply each term by the wave function. The constant term \(C\) cancels out.
\[\frac{\hbar^2}{2m}k^2\Psi(x, t) + V(x, t)\Psi(x, t) = \hbar\omega\Psi(x, t)\]
Make the equation a differential equation using the complex form of the wave equation.
\[ik = \frac{\partial}{\partial x} \quad k^2 = -\frac{\partial^2}{\partial x^2}\]
\[-i\omega = \frac{\partial}{\partial t} \quad \omega = i\frac{\partial}{\partial x}\]
This gives a complex differential equation.
\[-\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x, t)}{\partial x^2} + V(x, t)\Psi(x, t) = i\hbar\frac{\partial\Psi(x, t)}{\partial t}\]
Generalise to three dimensions using the laplacian \(\nabla\).
\[\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\]
\[-\frac{\hbar^2}{2m}\nabla^2\Psi(x, t) + V(x, t)\Psi(x, t) = i\hbar\frac{\partial\Psi(x, t)}{\partial t}\]
If the potential is made time independent \(V(x, t) \rightarrow V(x)\), the the wave function can be split into space and time dependent components using separation of variables \(\Psi(x, t) = \psi(x)f(t)\).
Substitute into the Schrödinger equation.
\[-\frac{\hbar^2}{2m}\nabla^2\psi(x)f(t) + V(x)\psi(x)f(t) = i\hbar\psi(x)f'(t)\]
This can be split into space and time dependent parts. As these are independent of each other they must both equal a constant which is the energy \(E\).
\[-\frac{\hbar^2}{2m}\frac{\nabla^2\psi(x)}{\psi(x)} + V(x) = i\hbar\frac{f'(t)}{f(t)} = E\]
The time component.
\[f'(t) = \frac{iE}{\hbar}f(t)\]
This can be integrated with \(C\) as an arbitrary constant of integration.
\[f(t) = Ce^{\frac{iEt}{\hbar}}\]
The rest is the time independent Schrödinger equation.
\[-\frac{\hbar^2}{2m}\nabla^2\psi(x) + V(x)\psi(x) = E\psi(x)\]