KAHIBARO
Table of Contents
A New Kind of Equation for Matter Waves
In classical mechanics, the motion of a particle is described by position, velocity, and force. In quantum mechanics, the central object is the wave function, usually written as $\psi$. The Schrödinger equation tells us how this wave function changes in space and time.
This equation plays a role similar to Newton's second law in classical physics. Newton's law predicts motion when forces are known. The Schrödinger equation predicts the evolution of the quantum state when the energy conditions of the system are known.
Why It Is Needed
Matter at very small scales does not behave like tiny classical balls. Electrons, atoms, and other microscopic objects show wave-like behavior. A wave description is therefore needed. The Schrödinger equation is the basic mathematical rule that governs these matter waves.
It does not directly give a definite path for a particle. Instead, it gives the wave function, from which measurable predictions can be extracted. The details of probability interpretation belong elsewhere, but here it is enough to say that $\psi$ contains the full quantum information about the system.
The Time-Dependent Schrödinger Equation
For one particle moving in one dimension, the time-dependent Schrödinger equation is
$$
i\hbar \frac{\partial \psi(x,t)}{\partial t}
=
-\frac{\hbar^2}{2m}\frac{\partial^2 \psi(x,t)}{\partial x^2}
+
V(x,t)\psi(x,t)
$$
Here, $m$ is the particle mass, $V(x,t)$ is the potential energy function, and $\hbar$ is the reduced Planck constant.
The left side describes how the wave function changes with time. The right side contains two parts. One part is related to kinetic energy, and the other is related to potential energy.
The time-dependent Schrödinger equation in one dimension is
$$
i\hbar \frac{\partial \psi}{\partial t}
=
-\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}
+
V\psi
$$
This is the fundamental evolution equation of nonrelativistic quantum mechanics.
Meaning of the Terms
The term
$$
-\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}
$$
is the kinetic part. The second spatial derivative measures how curved the wave function is in space. Greater curvature is associated with greater momentum variation and kinetic energy.
The term
$$
V(x,t)\psi(x,t)
$$
is the potential part. It tells how the environment, such as electric attraction or a confining wall, influences the wave.
The factor $i$, the imaginary unit, is essential. It allows oscillatory time evolution and is one reason quantum wave functions are usually complex.
Three-Dimensional Form
For a particle in three dimensions, the equation becomes
$$
i\hbar \frac{\partial \psi(\mathbf{r},t)}{\partial t}
=
-\frac{\hbar^2}{2m}\nabla^2 \psi(\mathbf{r},t)
+
V(\mathbf{r},t)\psi(\mathbf{r},t)
$$
Here, $\mathbf{r}$ is the position vector, and $\nabla^2$ is the Laplacian operator. In Cartesian coordinates,
$$
\nabla^2
=
\frac{\partial^2}{\partial x^2}
+
\frac{\partial^2}{\partial y^2}
+
\frac{\partial^2}{\partial z^2}
$$
So the equation includes curvature in all three spatial directions.
Relation to Energy
The Schrödinger equation reflects the idea that the total energy is the sum of kinetic and potential energy. In classical mechanics,
$$
E = \frac{p^2}{2m} + V
$$
In quantum mechanics, this becomes an equation involving operators acting on the wave function. Without going deeply into operator theory, the important replacements are
$$
E \to i\hbar \frac{\partial}{\partial t}
$$
and
$$
p \to -i\hbar \frac{\partial}{\partial x}
$$
in one dimension. Substituting these into the classical energy relation leads to the Schrödinger equation.
A useful way to remember the Schrödinger equation is to start from
$$
E = \frac{p^2}{2m} + V
$$
and replace energy and momentum by quantum differential operators.
The Time-Independent Schrödinger Equation
If the potential does not depend on time, it is often possible to separate the wave function into a space part and a time part:
$$
\psi(x,t) = \phi(x)T(t)
$$
This leads to the time-independent Schrödinger equation:
$$
-\frac{\hbar^2}{2m}\frac{d^2 \phi(x)}{dx^2} + V(x)\phi(x) = E\phi(x)
$$
In three dimensions,
$$
-\frac{\hbar^2}{2m}\nabla^2 \phi(\mathbf{r}) + V(\mathbf{r})\phi(\mathbf{r}) = E\phi(\mathbf{r})
$$
This equation is an eigenvalue equation. The allowed energies $E$ come from solving it together with the correct boundary conditions.
Why the Time-Independent Form Matters
The time-independent form is especially important for systems with stable energy values, such as electrons in atoms or particles trapped in a box. Instead of following the full time evolution directly, we solve for special states with definite energy.
These special solutions are called stationary states. Their time dependence has the form
$$
e^{-iEt/\hbar}
$$
so the full wave function is
$$
\psi(x,t) = \phi(x)e^{-iEt/\hbar}
$$
Although the wave function changes with time through this phase factor, the measurable probability pattern for a single stationary state does not change with time.
Stationary States and Quantized Energy
One of the most important consequences of the Schrödinger equation is that energies are often quantized. This means that only certain discrete energy values are allowed.
This does not happen because someone inserts discrete energies by hand. It appears naturally when the equation is solved with physical requirements such as continuity, finiteness, and boundary conditions.
For example, if a particle is confined to a region of space, only certain wave patterns fit properly. This is similar to standing waves on a string, where only certain wavelengths are allowed.
For bound systems, solving the time-independent Schrödinger equation often gives discrete allowed energies.
This is the origin of quantum energy levels.
General Structure of a Solution
A quantum problem based on the Schrödinger equation usually follows this pattern.
First, choose the potential energy function $V$. Second, write the Schrödinger equation. Third, solve for the wave function using boundary conditions. Fourth, identify the allowed energies and corresponding states.
The potential function determines the physics of the problem. Different choices of $V$ produce very different systems.
| Potential type | Physical meaning | Typical result |
|---|---|---|
| $V=0$ | Free particle | Traveling wave solutions |
| Infinite walls | Confined particle | Discrete energy levels |
| Harmonic form | Oscillator | Equally spaced energy pattern |
| Coulomb-like attraction | Atom-like system | Atomic energy levels |
The specific systems listed here are developed in their own chapters.
Boundary Conditions
Not every mathematical solution is physically acceptable. The wave function must satisfy appropriate conditions. In many basic problems, the wave function must be finite, continuous, and single-valued. In addition, it must fit the boundaries of the physical system.
For example, if a region is forbidden by an infinitely high barrier, the wave function must vanish there. If a particle is confined between walls, the wave function must match those wall conditions.
These restrictions are what lead to a limited set of acceptable solutions.
Linearity and Superposition
An important mathematical property of the Schrödinger equation is that it is linear. If $\psi_1$ and $\psi_2$ are solutions, then any combination
$$
\psi = c_1\psi_1 + c_2\psi_2
$$
is also a solution, where $c_1$ and $c_2$ are constants.
This is the mathematical basis of superposition in quantum mechanics. A system can be described by a combination of different possible states.
The Schrödinger equation is linear.
If $\psi_1$ and $\psi_2$ are solutions, then
$$
c_1\psi_1 + c_2\psi_2
$$
is also a solution.
A Visual Picture
The wave function can be imagined as a shape spread over space, and the Schrödinger equation tells how that shape bends and evolves. The potential influences this evolution. A deep well can trap the wave. A flat region can allow it to travel. A barrier can reduce or reshape it.
In this sketch, the black shape represents a simple potential well, and the blue curve represents a possible stationary-state wave function inside it.
Comparison of the Two Forms
| Form | Equation | Use |
|---|---|---|
| Time-dependent | $i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi$ | Describes full time evolution |
| Time-independent | $-\frac{\hbar^2}{2m}\nabla^2\phi + V\phi = E\phi$ | Finds energy states when $V$ is time independent |
The time-dependent equation is the more general one. The time-independent equation is a special and very useful case.
Limits of the Schrödinger Equation
The Schrödinger equation is a nonrelativistic equation. It works very well for many atoms, molecules, and low-speed particles. However, when particles move near the speed of light, or when particle creation and annihilation become important, more advanced theories are needed.
So the Schrödinger equation is fundamental, but it is not the final theory for every situation.
Key Ideas to Remember
The Schrödinger equation is the central equation of nonrelativistic quantum mechanics. It governs how the wave function changes with time. Its time-independent form is used to find allowed energy states. Boundary conditions determine which solutions are physically allowed. Because the equation is linear, superposition is possible. Quantized energy levels arise naturally from acceptable solutions in bound systems.
Essential formulas:
Time-dependent Schrödinger equation:
$$
i\hbar \frac{\partial \psi}{\partial t}
=
-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi
$$
Time-independent Schrödinger equation:
$$
-\frac{\hbar^2}{2m}\nabla^2\phi + V\phi = E\phi
$$
Stationary-state form:
$$
\psi(\mathbf{r},t) = \phi(\mathbf{r})e^{-iEt/\hbar}
$$