The Idea of a Wave Equation

By the late 1920s, experiment had established two uncomfortable facts. First, matter has a wave nature: de Broglie's relation $\lambda = h/p$ connects momentum to wavelength. Second, that wave-like quantity — which we call the wave function $\psi$ — must somehow encode all physically accessible information about a particle. The natural next question is: what equation does $\psi$ obey?

What we need from such an equation

A good wave equation for quantum mechanics should satisfy several requirements that we can read off from what we already know:

It must be linear. Superposition is observed — if $\psi_1$ and $\psi_2$ are both valid states, then $c_1 \psi_1 + c_2 \psi_2$ must also be a valid state. Only linear equations preserve this.
It must reproduce the de Broglie relation. A free particle with definite momentum $p$ and energy $E$ should correspond to a plane wave $\psi(x,t) = A\,e^{i(kx - \omega t)},$ where $k = p/\hbar$ and $\omega = E/\hbar$ .
It must encode the energy–momentum relation. For a non-relativistic particle of mass $m$ moving in a potential $V(x,t)$ , the classical energy relation is $E = \frac{p^2}{2m} + V.$ The equation for $\psi$ must be consistent with this.
It should be first-order in time. Knowing $\psi$ at one moment should determine $\psi$ at all future moments — just as Newton's second law (first-order in velocity) determines the full trajectory once position and velocity are specified.

Reading off the operators

The plane-wave form $\psi = A\,e^{i(kx-\omega t)}$ lets us translate derivatives into physical quantities. Differentiating once with respect to $x$ brings down $ik = ip/\hbar$ :

$\frac{\partial \psi}{\partial x} = \frac{ip}{\hbar}\,\psi \quad \Longrightarrow \quad p \leftrightarrow -i\hbar\frac{\partial}{\partial x}.$

Differentiating twice with respect to $x$ brings down $-k^2 = -p^2/\hbar^2$ :

$\frac{\partial^2 \psi}{\partial x^2} = -\frac{p^2}{\hbar^2}\,\psi \quad \Longrightarrow \quad p^2 \leftrightarrow -\hbar^2\frac{\partial^2}{\partial x^2}.$

Differentiating once with respect to $t$ brings down $-i\omega = -iE/\hbar$ :

$\frac{\partial \psi}{\partial t} = -\frac{iE}{\hbar}\,\psi \quad \Longrightarrow \quad E \leftrightarrow i\hbar\frac{\partial}{\partial t}.$

These correspondences are sometimes called operator substitutions: whenever $E$ or $p$ appears in a classical expression, replace it with the matching differential operator acting on $\psi$ .

Assembling the equation

Applying both substitutions to the classical energy relation $E = p^2/(2m) + V$ and letting the resulting operators act on $\psi$ :

$i\hbar\frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + V\psi.$

This is the time-dependent Schrödinger equation in one dimension. Notice:

The left side encodes energy through the time derivative.
The first term on the right is the kinetic energy contribution.
The second term $V\psi$ is the potential energy contribution.
The equation is indeed first-order in $t$ and linear in $\psi$ — exactly as required.

The wave function carries probability, not energy

One might ask: is $\psi$ itself a physical wave like a water wave or an electromagnetic field? The answer is no — $|\psi(x,t)|^2$ is a probability density. If the particle is in state $\psi$ , the probability of finding it between $x$ and $x + dx$ at time $t$ is

$P(x,t)\,dx = |\psi(x,t)|^2\,dx.$

For this to make sense globally, the wave function must be normalizable:

$\int_{-\infty}^{\infty} |\psi(x,t)|^2\,dx = 1.$

Requiring normalization is not a separate postulate — once $\psi$ satisfies the Schrödinger equation with a square-integrable initial condition and a real potential $V$ , the norm is conserved for all time. This can be shown by differentiating the integral with respect to $t$ and substituting the equation.

Looking ahead

The Schrödinger equation in full will be explored in the next lesson. The key takeaway here is how the equation is motivated: linearity, the de Broglie relation, the classical energy–momentum connection, and first-order time evolution together point almost uniquely to its form. Understanding this logic makes the equation feel inevitable rather than mysterious.