Schrödinger Equation

In one sentence: The Schrödinger equation says how a quantum particle’s wave changes over time, the way Newton’s laws say how a moving ball changes its position.

$i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi$

Read it out loud: “i times h-bar times the partial derivative of psi with respect to t equals H-hat times psi. Psi is the wave function, t is time, and H-hat is the energy operator.”

What each symbol means

Symbol	Meaning
$i$	The imaginary unit, with $i^2 = -1$ . It is what makes the phase of the wave function turn as time passes.
$\hbar$	The reduced Planck constant, $1.054571817 \times 10^{-34}$ J s (joule seconds).
$\frac{\partial \psi}{\partial t}$	How fast the wave function changes at one fixed point as time passes. The partial derivative means only time is changing here.
$\psi$	The wave function, a complex number at each position and time. Its squared size gives a probability density (see the Born rule).
$\hat{H}$	The Hamiltonian, the energy operator. For one particle of mass $m$ in a potential $V$ , $\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V$ . Its result is in joules (J).
$t$	Time, in seconds (s).
$m$	The mass of the particle, in kilograms (kg).
$\nabla^2$	The Laplacian, the sum of second spatial derivatives. In one dimension it is $\partial^2 / \partial x^2$ , in units of 1/m^2.
$V$	The potential energy at each position, in joules (J). It is the landscape the particle moves through.
$E$	The energy of a state, in joules (J), in the time-independent form $\hat{H}\psi = E\psi$ .

What the formula is doing

The Schrödinger equation is the basic equation for how a quantum state changes with time. It does for the wave function what Newton’s second law does for motion. If you know the state now and the potential acting on the particle, it predicts the state later.

Read the left side as how fast the wave changes at each point, and the right side as applying the energy operator to the wave. The operator $\hat{H}$ has two parts: $-\frac{\hbar^2}{2m}\nabla^2$ , which describes motion, and $V$ , which describes the potential energy at each position. The factor $i$ makes the phase of the wave turn as time passes.

When the energy is fixed, the time-independent form $\hat{H}\psi = E\psi$ applies. Its solutions are the states with a definite energy $E$ . Those states keep the same probability pattern as time passes, which is why they are called stationary states.

Solving the equation means finding wave functions that obey it and the conditions of the situation, such as walls where the wave must vanish. Only certain energies pass that test. That is why a particle that is trapped has discrete energy levels, as the box and hydrogen examples show.

The equation is linear. If two solutions exist, their sum is also a solution. That is why a particle can be in a combination of several energy states, and why the energy of such a combination is not fixed.

An everyday picture

Think of a guitar string clamped at both ends. Only certain wave patterns fit between the clamps, so only certain pitches can sound. A particle trapped in a box is like that string: only certain energy patterns fit inside the walls, and each one has its own energy.

Worked example: An electron in a 1 nm box

An electron (mass $m = 9.1093837 \times 10^{-31}$ kg) is trapped in a one-dimensional box of width $L = 1.0$ nm $= 1.0 \times 10^{-9}$ m, with walls too high to escape. Its allowed energies are $E_n = \frac{n^2 h^2}{8 m L^2}$ for $n = 1, 2, 3, \dots$ , with $h = 6.62607015 \times 10^{-34}$ J s.

Compute the energy for $n = 1$ : $E_1 = \frac{h^2}{8 m L^2} = \frac{(6.62607015 \times 10^{-34})^2}{8 (9.1093837 \times 10^{-31})(1.0 \times 10^{-9})^2} \approx 6.02 \times 10^{-20}$ J.
Convert to electron volts: $E_1 = \frac{6.02 \times 10^{-20}}{1.602 \times 10^{-19}} \approx 0.376$ eV.
Next level, $n = 2$ : $E_2 = 2^2 E_1 = 4 E_1 \approx 1.50$ eV.
Level $n = 3$ : $E_3 = 3^2 E_1 = 9 E_1 \approx 3.38$ eV.

Answer: In this box the electron can have energies of 0.376 eV, 1.50 eV, 3.38 eV and so on. The lowest energy, 0.376 eV, is not zero, because confining a particle always costs some energy.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Forgetting that the box width is in meters. Convert 1 nm to $1.0 \times 10^{-9}$ m before squaring. The $L^2$ in the denominator is what makes these energies small.
Believing the lowest energy is zero. The lowest level is $n = 1$ , and it is above zero because confining a particle costs energy.
Using $n = 0$ . For this box the quantum number starts at 1, because $n = 0$ would mean no wave at all.
Treating $\hat{H}\psi = E\psi$ as the whole story. That form describes states of fixed energy. The full time-dependent equation is needed to see how a state changes in time.

Where you will meet it

Chemistry, where solutions for the hydrogen atom and other atoms give orbital shapes and energy levels.
Quantum dots and semiconductor design, where electrons trapped in tiny regions follow box-like energy levels.
Tunneling calculations, where solutions of the equation predict how electrons cross energy barriers.
Quantum chemistry software, which solves the equation approximately to predict the energy levels of molecules.

Questions people ask

Is the Schrödinger equation the same as Newton's law?

They play the same kind of role: given a starting state, they predict what comes next. Newton’s law tracks position and speed, while the Schrödinger equation tracks the wave function, which gives probabilities.

Where does the i come from?

The factor $i$ makes the wave evolve by turning its phase over time. This keeps the total probability constant, which is what a physical theory needs.

Can I solve it by hand?

Only for simple cases such as the box, the harmonic oscillator and the hydrogen atom. For most real systems, computers solve it approximately.

Born Rule: The Born rule says the chance of finding a particle at a spot, or of getting an outcome, is the squared size of its wave at that spot.
De Broglie Wavelength: Every moving object has a wavelength, and the heavier or faster it moves, the shorter that wavelength gets.
Hydrogen Energy Levels: A hydrogen electron can only sit on certain energy steps, and every jump between steps gives off or takes in light of one exact color.