KAHIBARO
Table of Contents
A New Kind of Description
In classical physics, an object is described by quantities such as position, velocity, and energy. In quantum mechanics, the state of a particle is described in a different way, by a wave function. The wave function is the central mathematical object of quantum theory.
A wave function is usually written as $\psi$. It contains the information needed to describe the quantum state of a system. For a single particle moving in one dimension, the wave function can be written as $\psi(x,t)$, which means it depends on position $x$ and time $t$.
Unlike the position of a classical particle, the wave function is not something we directly observe. Instead, it is a mathematical function that helps us predict the outcomes of measurements.
The wave function $\psi(x,t)$ is not itself a directly measurable physical quantity. It is a mathematical description of a quantum state.
Dependence on Position and Time
The notation $\psi(x,t)$ tells us that the wave function changes from place to place and can also change with time. At each position $x$ and time $t$, the wave function has a value.
In general, the wave function can be a complex number. This means it may contain both a real part and an imaginary part. Complex numbers are important in quantum mechanics because they allow wave behavior such as interference to be described naturally.
A simple example of a wave-like function is
$$
\psi(x,t) = A e^{i(kx - \omega t)}
$$
where $A$ is the amplitude, $k$ is the wave number, and $\omega$ is the angular frequency. This form looks like an ordinary traveling wave, but in quantum mechanics it represents a quantum state.
Why It Is Called a Wave Function
The name comes from the fact that quantum particles show wave-like behavior. Electrons, for example, can produce interference patterns. To describe this, quantum mechanics uses a function that behaves mathematically like a wave.
This does not mean the particle is literally a tiny water wave. It means the mathematical description of the particle has wave properties. Because of this, different parts of a wave function can add together or cancel out, just as ordinary waves do.
Visualizing a Wave Function
A wave function can be drawn as a curve showing how $\psi$ changes with position. Since $\psi$ may be complex, simple sketches often show only its real part, or sometimes just its magnitude.
This picture is only a rough illustration. In real quantum problems, the wave function may have more complicated shapes.
Wave Function and the State of a System
The wave function represents the state of a quantum system. If you know the wave function, then in principle you know everything that quantum mechanics allows you to know about that system at that time.
For a single particle in one dimension, the wave function depends on one position variable. For a particle in three dimensions, it becomes $\psi(x,y,z,t)$. For systems with more than one particle, the wave function depends on the coordinates of all particles.
The table below shows some common forms.
| System | Wave function form |
|---|---|
| One particle in one dimension | $\psi(x,t)$ |
| One particle in three dimensions | $\psi(x,y,z,t)$ |
| Two particles in one dimension | $\psi(x_1,x_2,t)$ |
| Two particles in three dimensions | $\psi(\mathbf{r}_1,\mathbf{r}_2,t)$ |
Acceptable Wave Functions
Not every mathematical function can be a physical wave function. A physically acceptable wave function should satisfy some basic conditions.
It should be single-valued, so that at each point in space and time it has only one value. It should be finite, so it does not become infinitely large at ordinary points. It should usually be continuous, and in many situations its slope is also continuous.
These conditions help make the wave function physically meaningful and mathematically usable.
A physical wave function should be single-valued and finite, and in ordinary situations it should be continuous.
Real and Complex Forms
A wave function can be real, complex, or partly both. For example,
$$
\psi(x) = A \sin(kx)
$$
is a real wave function, while
$$
\psi(x) = A e^{ikx}
$$
is complex.
Even when the wave function is complex, the measurable predictions of quantum mechanics come from combinations of $\psi$ and its complex conjugate $\psi^*$. This is why complex wave functions are not a problem physically.
Standing Waves and Bound States
Some quantum systems have wave functions that look like standing waves. This often happens for particles confined to a region of space. The wave function can have nodes, points where its value is zero, and peaks where it is larger.
These shapes are important because only certain standing-wave patterns may fit inside a confined region. This is connected with quantized energy levels, which are treated in other chapters.
Wave Function and Matter Waves
The idea of the wave function is closely related to matter waves. If particles such as electrons have wave-like properties, then a wave function gives a mathematical description of those properties.
A wave function can spread out through space, meaning the quantum state is not concentrated at a single point in the classical sense. This is one of the major differences between classical and quantum descriptions.
Superposition of Wave Functions
Because wave functions behave like waves, they can be added together. If $\psi_1$ and $\psi_2$ are possible wave functions, then another possible wave function can often be written as
$$
\psi = c_1 \psi_1 + c_2 \psi_2
$$
where $c_1$ and $c_2$ are constants, often complex.
This is called superposition. It is one of the most important features of quantum mechanics. It leads to interference effects and many distinctly quantum phenomena.
If $\psi_1$ and $\psi_2$ are allowed quantum states, then a linear combination such as $c_1\psi_1 + c_2\psi_2$ is also an allowed state in many quantum systems.
What the Wave Function Does Not Mean
It is important not to confuse the wave function with an ordinary physical wave in space, like a vibrating string. The wave function is not a direct picture of a material object moving up and down.
Also, the value of $\psi$ itself is usually not what an experiment measures. The connection between the wave function and measurable probabilities is a separate topic, treated in the probability interpretation chapter.
Summary View
The wave function is the mathematical description of a quantum state. It is usually written as $\psi$, and for a particle it depends on position and time, such as $\psi(x,t)$ or $\psi(x,y,z,t)$. It can be complex, it behaves mathematically like a wave, and it allows superposition. A physically acceptable wave function should be single-valued, finite, and generally continuous.
Key idea: the wave function $\psi$ describes the quantum state of a system and is the starting point for predicting all quantum behavior.