Schrödinger Equation

The Schrödinger equation tells you how a quantum state changes in nonrelativistic quantum mechanics. If you know the wave function $\psi$ and the potential energy $V$, this equation tells you how $\psi$ evolves and which energy states are allowed.

For one particle in three dimensions, the time-dependent equation is commonly written as

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

Here $m$ is the particle mass, $V$ is the potential energy, and $\hbar$ is the reduced Planck constant. This is the standard starting point when a nonrelativistic model is appropriate, meaning relativistic effects are small enough to ignore.

What The Schrödinger Equation Means

The equation links two ideas: how the wave function changes in time and how the system's energy acts on that wave function. The $\nabla^2$ term is tied to kinetic energy, while $V$ represents potential energy.

You should not read $\psi$ as a classical wave like the height of water. In the standard interpretation, the measurable quantity is $|\psi|^2$, which gives a probability density after normalization.

That is the key shift from classical mechanics. The equation does not usually predict one exact path for a particle. It predicts how the probability structure of the system changes.

When The Time-Independent Form Applies

The time-dependent Schrödinger equation is the general form. A second form appears only when the potential does not depend on time and you are looking for stationary states with definite energy.

In one dimension, that time-independent form is

$$-\frac{\hbar^2}{2m}\frac{d^2 \psi}{dx^2} + V(x)\psi = E\psi$$

This is not a different law. It is a special case of the time-dependent equation under those conditions. If the potential changes with time, you should not expect this simpler form to describe the full situation.

Worked Example: Particle In A One-Dimensional Box

A standard example is a particle confined between $x=0$ and $x=L$, with potential energy

$$V(x) = \begin{cases} 0, & 0 < x < L \\ \infty, & \text{outside the box} \end{cases}$$

Inside the box, the potential is zero, so the time-independent equation becomes

$$-\frac{\hbar^2}{2m}\frac{d^2 \psi}{dx^2} = E\psi$$

At the walls, the wave function must vanish:

$$\psi(0) = 0, \qquad \psi(L) = 0$$

Those boundary conditions remove most mathematical solutions and leave only certain stationary states:

$$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right), \qquad n = 1,2,3,\dots$$

and the allowed energies are

$$E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$

This is the main idea to remember. The equation alone is not enough; the boundary conditions matter too. Together, they allow only a discrete set of energies rather than every possible value.

If the box gets larger, the allowed energies get smaller because $E_n \propto 1/L^2$. If the box gets smaller, the energy levels spread farther apart.

Why This Example Makes Quantum Mechanics Click

The particle-in-a-box model is simple, but it makes one quantum idea clear very quickly: confinement can produce quantized energy. That same pattern appears more broadly in atoms, quantum wells, and other bound systems.

It also shows why boundary conditions are not a side detail. In quantum mechanics, the physical setup and the allowed wave functions are tightly linked.

Common Schrödinger Equation Mistakes

• Treating $\psi$ itself as a probability. In the standard interpretation, the probability density is $|\psi|^2$ after normalization.
• Using the time-independent equation as if it always applies. It is the right tool only for stationary-state problems with time-independent potential.
• Expecting the equation to give an exact classical trajectory. In general, it evolves a wave function, not a single path.
• Forgetting that boundary conditions can change which solutions are physically allowed.

Where The Schrödinger Equation Is Used

It is used across atomic physics, molecular physics, tunneling problems, semiconductor models, and many parts of quantum chemistry. In each case, the exact potential and the system details change, but the same core framework remains.

For very high speeds or when relativistic effects matter, the Schrödinger equation is not the complete model. In that regime, more advanced equations are needed.

Try A Similar Change

Keep the same box but replace $L$ with $2L$. Without doing much algebra, predict what happens to $E_1$ and the spacing between nearby energy levels. If you want a useful comparison after that, look at the wave equation and notice how both equations connect differential equations to physical constraints, but in different ways.