# Schrödinger Equation

> The Schrödinger equation says how a quantum particle's wave changes from moment to moment, the way Newton's laws say how a moving ball changes its position.

Page: https://formulabro.yunusemrevurgun.com/quantum-mechanics/schrodinger-equation/  
Category: Quantum Mechanics  
Published: 2026-10-22  
Updated: 2026-10-22

## Formula

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

Read it out loud: "i times h-bar times the partial derivative of psi with respect to t equals H-hat times psi. The letter i is the imaginary unit, which is the number whose square is minus one. H-bar is the reduced Planck constant, a tiny fixed number. The curly d, called partial, means how fast psi changes over time at one fixed point. Psi, said sigh, is the wave function. T is time. H-hat is the energy operator, the rule that finds the energy effect of the wave."

## Before you start

- **Powers of ten, like $10^{-34}$ and $10^{-9}$**: Physics numbers are often too big or too small to write out, so they are written as a short number times a power of ten. The small raised number, the **exponent**, counts the multiplications by 10, and a minus sign means divide instead. So $10^{-34}$ is a decimal point followed by 33 zeros and then a 1. One nanometer is $10^{-9}$ m, a billionth of a meter. When you multiply powers of ten, add the exponents, and when you divide, subtract them. This is called **scientific notation**.
- **The imaginary unit $i$**: The **imaginary unit** $i$ is defined by $i^2 = -1$. No ordinary positive or negative number squared gives a negative answer, so $i$ belongs to a wider kind of number called a **complex number**. A complex number has a size and a direction, and it can be drawn as an arrow. Multiplying by $i$ turns the arrow a quarter turn, 90 degrees. In the Schrödinger equation, the $i$ makes the wave turn its direction as time passes.
- **Partial derivative and the rate of change, $\frac{\partial}{\partial t}$**: A **derivative** measures how fast something changes. If a car travels 60 km in one hour, its rate of change of distance is 60 km per hour. The curly symbol $\partial$ is read 'partial'. It means only one input is allowed to change while the others are held fixed. So $\frac{\partial \psi}{\partial t}$ is how fast the wave changes over time at one fixed position, with the position kept still.
- **Operator and the hat symbol, $\hat{H}$**: An **operator** is a rule that takes one wave and turns it into another. The hat symbol, written over a letter like $\hat{H}$, marks an operator. So $\hat{H}\psi$ means apply the rule $\hat{H}$ to the wave $\psi$. The operator $\hat{H}$ is the energy rule, which says how the wave's energy is found at each point.
- **Potential energy $V$ and the Hamiltonian**: **Potential energy** is stored energy that depends on where something is, like a ball held high above the ground. The letter $V$ stands for the potential energy at each position, the landscape the particle moves through. The **Hamiltonian**, written $\hat{H}$ and named after the mathematician William Rowan Hamilton, adds the energy of motion to the potential energy. It is the energy rule for the whole system.
- **The Laplacian $\nabla^2$ and second derivatives**: A **second derivative** is a derivative of a derivative. It measures how sharply a curve bends. The Laplacian, written $\nabla^2$ and read 'nabla squared', adds up these bending measures in every direction of space. In one dimension it is the second derivative with respect to position, which measures the bending along one line.

More notation help: https://formulabro.yunusemrevurgun.com/math-basics/

## What each symbol means

- $i$: The **imaginary unit**, with $i^2 = -1$. It has no size on the ordinary number line, and it is what makes the phase of the wave turn as time passes. It is a pure number with no unit.
- $\hbar$: The **reduced Planck constant**: $1.054571817 \times 10^{-34}$ J s, with the unit joule seconds, meaning energy multiplied by time. It is tiny, so quantum effects matter only for very small things. Everyday objects are so large that their quantum behavior is invisible.
- $\frac{\partial \psi}{\partial t}$: How fast the wave function changes at one fixed point as time passes. It is a rate, so it is measured in the wave's own units per second. The partial derivative means that only time is changing in this term, while position is held fixed.
- $\psi$: The **wave function**, a complex number at each position and time. Its unit in one dimension is one over the square root of a meter, written $\text{m}^{-1/2}$. Its squared size gives a probability density, as explained by the Born rule.
- $\hat{H}$: The **Hamiltonian**, the energy operator. For one particle of mass $m$ moving in a potential $V$, it equals $-\frac{\hbar^2}{2m}\nabla^2 + V$. Applied to a wave, it gives the energy effect of that wave, and its result has the units of energy times the wave function.
- $t$: **Time**, measured in seconds (s). It is the clock that the wave follows, and the equation tells you how the wave at one moment leads to the wave at the next.
- $m$: The **mass** of the particle, in kilograms (kg). An electron has a mass of $9.1 \times 10^{-31}$ kg. A heavier particle has a smaller second term in the energy rule, so its wave spreads more slowly.
- $\nabla^2$: The **Laplacian**, the sum of the second spatial derivatives, which measures how sharply the wave bends in space. In one dimension it is $\partial^2 / \partial x^2$, with units of one over meters squared, written 1/m$^2$.
- $V$: The **potential energy** at each position, in joules (J). It is the landscape the particle moves through, like hills and valleys. Where $V$ is high, the particle has less energy left for motion.
- $E$: The **energy of a state**, in joules (J). It appears in the time-independent form $\hat{H}\psi = E\psi$, for a state with one definite energy. Each allowed energy has its own value of $E$, and the wave for that state keeps the same probability pattern as time passes.

## The formula piece by piece

- $\frac{\partial \psi}{\partial t}$: Start with the left side. It is the rate at which the wave changes over time at one fixed point. This is a rate, so it is measured per second, and it describes how the wave is moving forward in time.
- $i \hbar$: Multiply that rate by $i$ and $\hbar$. The factor $i\hbar$ means the change is a turning motion in the complex plane, not a simple growing or shrinking. The number $\hbar$ sets how quickly that turning happens for a given energy.
- $\hat{H}$: The energy rule for one particle. It is built from two pieces: $-\frac{\hbar^2}{2m}\nabla^2$, which describes the energy of motion through the way the wave bends, and $V$, which describes the potential energy at each position.
- $\hat{H} \psi$: Apply the energy rule to the wave. The result is another wave, which at each point gives the energy effect of the original wave. So the equation compares two waves, the wave's change in time and its energy effect.
- $i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi$: The whole formula in one sentence: **the wave's rate of change in time, multiplied by $i$ and the reduced Planck constant, equals what the energy rule does to the wave.** Knowing the wave now and the energy rule tells you the wave later.
- $\hat{H} \psi = E \psi$: The time-independent form, for a state with one definite energy $E$. Applying the energy rule to the wave gives the same wave multiplied by a number, $E$. Such a wave keeps the same probability pattern over time, which is why it is called stationary.

## What the formula is doing, and why

**What goes in.** The wave function $\psi$ at one moment, along with the potential $V$ that the particle moves through. **What happens.** The equation sets how fast the wave changes at each point, using the energy rule $\hat{H}$. **What comes out.** The wave a little later. Repeating the step predicts the wave at any later time, in the same way that Newton's second law predicts a ball's later position from where it is now.

**Why the factor $i$?** The $i$ makes the wave turn its direction in the complex plane as time passes, instead of making it grow or shrink. This keeps the total probability at exactly 1, the same before and after. That property is built into the equation, and it is one reason the wave changes in such a regular way.

**Why does the energy rule have two parts?** The term $-\frac{\hbar^2}{2m}\nabla^2$ describes how the wave bends and spreads, which corresponds to the energy of motion. The term $V$ describes the energy that depends on position. Adding them gives the total energy rule. For a heavy particle the first term is small, so its wave spreads slowly.

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

**Solutions and energy levels.** Solving the equation means finding waves that obey it and the conditions of the setup, such as walls where the wave must be zero. Only certain energies pass that test, which is why a trapped particle has discrete energy levels, as the box and hydrogen examples show. The equation is linear, so if two solutions exist, their sum is also a solution, and a particle can be in a mixture of energy states.

## 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, and each pattern on the string has its own pitch.

## Worked example: An electron in a 1 nm box

An electron, with mass $m = 9.1093837 \times 10^{-31}$ kg, is trapped in a one-dimensional box of width $L = 1.0$ nm, which is $1.0 \times 10^{-9}$ m, with walls too high to escape. Boxes this small appear in quantum dots, which are crystals a few nanometers across used in some bright TV screens. The 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.

1. **Write the box formula and say what each letter means.** The energy of level $n$ is $E_n = \frac{n^2 h^2}{8 m L^2}$. The number $n$ labels the level, and the energy grows with $n^2$. We start with the lowest level, $n = 1$, so the $n^2$ term equals 1 and drops out of the calculation.
2. **Square the width and the Planck constant.** The width is $L = 1.0 \times 10^{-9}$ m, so $L^2 = 1.0 \times 10^{-18}$ m$^2$. Squaring a power of ten doubles its exponent: $(10^{-9})^2 = 10^{-18}$. Squaring the Planck constant gives $h^2 = (6.62607015 \times 10^{-34})^2 \approx 4.3905 \times 10^{-67}$ J$^2$ s$^2$, since $6.626^2 \approx 43.9$ and the exponent doubles from $-34$ to $-68$.
3. **Compute the bottom of the fraction.** The denominator is $8 m L^2 = 8 \times 9.1093837 \times 10^{-31} \times 1.0 \times 10^{-18}$. Multiply the front numbers, $8 \times 9.1093837 = 72.875$. Add the exponents, $-31 + (-18) = -49$, which gives $72.875 \times 10^{-49}$. Moving the decimal point one place gives $7.2875 \times 10^{-48}$ kg m$^2$.
4. **Divide to get the energy in joules.** $E_1 = \frac{4.3905 \times 10^{-67}}{7.2875 \times 10^{-48}}$. Divide the front numbers, $4.3905 / 7.2875 \approx 0.60247$. Subtract the exponents, $-67 - (-48) = -19$, which gives $0.60247 \times 10^{-19}$ J. Moving the decimal point one place gives $E_1 \approx 6.0247 \times 10^{-20}$ J.
5. **Convert to electron volts.** One electron volt is $1.602176634 \times 10^{-19}$ J. Dividing gives $\frac{6.0247 \times 10^{-20}}{1.602176634 \times 10^{-19}} \approx 0.376$ eV, because the front numbers give 0.376 after the exponents are handled. Electron volts are the easier unit here, since the energy of the lowest level is then a number near one.
6. **Find the next two levels.** Level 2 has $n^2 = 4$ times the energy of level 1, so $E_2 = 4 \times 0.376 \approx 1.50$ eV. Level 3 has $n^2 = 9$ times the energy of level 1, so $E_3 = 9 \times 0.376 \approx 3.38$ eV. The levels spread farther apart as $n$ grows.

**Answer:** In this box the electron can have energies of about 0.376 eV, 1.50 eV, 3.38 eV, and so on. The lowest energy is not zero, because confining a particle always costs some energy. A trapped electron cannot sit with no energy at all, the way a ball can rest on the floor of a room.

## 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 so small.
- Believing the lowest energy is zero. The lowest level is $n = 1$, and it sits above zero, because confining a particle costs energy.
- Using $n = 0$. For this box the quantum number starts at 1, because $n = 0$ would give no wave at all.
- Treating $\hat{H}\psi = E\psi$ as the whole story. That form describes states of one 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 of the equation for the hydrogen atom and other atoms give the shapes of orbitals and their energy levels.
- Quantum dots and semiconductor design, where electrons trapped in tiny crystals follow box-like energy levels, which sets the color of the light they emit.
- Tunneling calculations, where solutions of the equation predict how particles pass through energy barriers they could not climb over in ordinary physics.
- 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 a similar role: given a starting state, both predict what comes next. Newton's second law tracks a ball's position and speed, while the Schrödinger equation tracks the wave function, which gives probabilities rather than one exact position.

### Where does the $i$ come from?

The factor $i$ makes the wave turn its phase over time instead of growing or shrinking. This keeps the total probability at 1, which is what a physical theory needs.

### Can I solve it by hand?

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

## Related formulas

- [Born Rule](https://formulabro.yunusemrevurgun.com/quantum-mechanics/born-rule/)
- [De Broglie Wavelength](https://formulabro.yunusemrevurgun.com/quantum-mechanics/de-broglie-wavelength/)
- [Hydrogen Energy Levels](https://formulabro.yunusemrevurgun.com/quantum-mechanics/hydrogen-energy-levels/)

## Sources

- [Schrödinger equation, Wikipedia](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation)
- [Particle in a box, Wikipedia](https://en.wikipedia.org/wiki/Particle_in_a_box)

From Formula Bro (https://formulabro.yunusemrevurgun.com/), a side project developed by yunusemrevurgun.com.