In quantum mechanics, wave function is a mathematical object used to describe a quantum particle. This wave function evolves according to the Schrödinger equation. For a single particle, the time-dependent 3-dimensional Schrödinger equation is stated as the following:

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

Meaning of symbols are:

• $i$ - Imaginary unit (A solution to the equation $i^2 + 1 = 0$)
• $\hbar$ - Reduced planck constant ($\hbar = h/2\pi$, with value $\hbar \approx 1.054 571 817\times 10^{-34}J\cdot s$)
• $\partial / \partial t$ - Derivative with respect to time
• $\nabla^2$ - Sum of second derivatives with respect to space ($\nabla^2 = \partial^2 / \partial x^2 + \partial^2 / \partial y^2 + \partial^2 / \partial z^2$)
• $\Psi$ - Particle wave function
• $m$ - Mass of the particle
• $V$ - Potential energy of the particle

Because quantum theory is linear, multiple particles combine their wave functions in a linear fashion (i.e. they just add up) and form a single unified wave function that describes the entire system. This combined wave function also evolves according to Schrödinger equation.

Note that Schrödinger equation is single-order with respect to time, but second-order with respect to space. This is unlike the classical wave equation describing (the equation describing sound and water waves), which is second-order both in time and space.

We start from the well-known equation of energy conservation: $$ E = E_K + V$$ Where $E$ is the total energy of a particle, $E_K$ is its kinetic energy and $V$ is its potential energy. Kinetic energy of a particle is defined as $E_K = mv^2 / 2$ while momentum of a particle is defined as $p = mv$. Solving the latter for $v$ and substituting it in $E_K$, then substituting $E_K$ into the equation for total energy $E$, we get the following equation: $$E = \frac{p^2}{2m} + V$$ We promote energy $E$ into an operator $\hat{E} = i\hbar \partial / \partial t$ and similarily we promote $p$ the momentum into an operator $\hat{p} = -i\hbar\nabla^2$: $$\hat{E}\Psi = - \frac{\hat{p}^2}{2m}\Psi + V\Psi$$ $$\boxed{i\hbar \frac{\partial\Psi}{\partial t} = \left(-\frac{\hbar^2}{2m}\nabla^2 + V\right)\Psi}$$ Thus, we directly derived the Schrödinger simply by promoting $E$ and $p$ to operators. But of course, this derivation can only be understood as informal.

We derived the Schrödinger from nonrelativistic energy-momentum relation $E = \frac{p^2}{2m} + V$ which makes the Schrödinger inherently nonrelativistic i.e. it cannot describe particles moving close to the speed of light nor particles with no mass (which always travel at the speed of light). We would have to have used the relativistic energy-momentum relation instead: $$E^2 = p^2c^2 + m^2c^4 $$ Promoting $E$ and $p$ to operators $\hat{E} = i\hbar \partial / \partial t$ and $\hat{p} = -i\hbar\nabla^2$ yields the Klein-Gordon equation $$\left(\frac{\partial^2\Psi}{\partial t^2} - c^2\nabla^2 + \frac{m^2c^4}{\hbar^2}\right)\Psi = 0$$ The issue with this equation is that, unlike Schrödinger, Klein-Gordon equation is a differential equation that is second-order in time.

It was Paul Dirac who correctly found a relativistic analogue of Schrödinger equation by postulating the existence of $\alpha, \beta, \gamma, \delta$ such that: $$E = \sqrt{p^2c^2 + m^2c^4} = \alpha p_x c + \beta p_y c + \gamma p_z c + \delta mc^2$$ As it turns out, $\alpha, \beta, \gamma, \delta$ cannot be ordinary complex numbers - they must be 4×4 complex matrices. Thus, wavefunction $\Psi$ in Dirac's equation is no longer a complex wave function but a 4-component Dirac spinor.

Still, Dirac's equation does not describe creation and anahilation of particles which we see e.g. electron-positron interaction. The theory further refined by promoting wave function itself to an operator acting on Fock space (combining spaces containing 0 particles, 1 particle, 2 particles, …).

This is called the second quantization, and it's the framework of quantum field theory (QFT). Also, in QFT, Klein-Gordon equation rises from the ashes successfuly being able to describe spin-0 partcles.