KAHIBARO
Table of Contents
From wave function to probability
In quantum mechanics, the wave function, usually written as $\psi$, does not directly tell us the exact position or exact motion of a particle in the classical sense. Instead, it tells us how to calculate probabilities. This is one of the most important changes from classical physics to quantum physics.
The probability interpretation says that the wave function is connected to the likelihood of finding a particle in different places or states when a measurement is made. The wave function itself can be positive, negative, or even complex, so it cannot by itself be a probability. A probability must be a real number greater than or equal to zero. For this reason, the physically meaningful quantity is the squared magnitude of the wave function, written as $|\psi|^2$.
The basic rule of the probability interpretation is
$$P \propto |\psi|^2$$
More precisely, the probability of finding a particle in a small region is given by $|\psi|^2$ times the size of that region.
Probability density
For a particle moving in one dimension, the quantity $|\psi(x,t)|^2$ is called the probability density. It tells us how probability is distributed along the $x$ axis at time $t$.
If we look at a very small interval between $x$ and $x + dx$, then the probability of finding the particle in that interval is
$$dP = |\psi(x,t)|^2 dx$$
This means that large values of $|\psi|^2$ correspond to places where the particle is more likely to be found, and small values correspond to places where it is less likely to be found.
For a finite interval from $a$ to $b$, the probability is
$$P(a \le x \le b) = \int_a^b |\psi(x,t)|^2 \, dx$$
In three dimensions, the idea is the same, but we use volume instead of length:
$$dP = |\psi(\mathbf{r},t)|^2 dV$$
and
$$P(\text{in region } R) = \int_R |\psi(\mathbf{r},t)|^2 \, dV$$
Why the square of the wave function
The wave function can be complex, for example
$$\psi = a + ib$$
where $a$ and $b$ are real numbers. A complex number cannot be used directly as a probability. But its squared magnitude is real and nonnegative:
$$|\psi|^2 = \psi^* \psi = a^2 + b^2$$
where $\psi^*$ is the complex conjugate of $\psi$.
This is why quantum mechanics uses $|\psi|^2$ rather than $\psi$ itself. The quantity $|\psi|^2$ behaves like a true probability density.
If $\psi$ is complex, the probability density is
$$|\psi|^2 = \psi^* \psi$$
not $\psi^2$.
Meaning of measurement
The probability interpretation is about measurement outcomes. Before measurement, quantum mechanics does not assign a single definite position to the particle in the classical way. Instead, it provides probabilities for different possible results.
If many identical systems are prepared in the same quantum state and the same measurement is repeated, the fraction of times a result appears approaches the probability predicted by the wave function. In this sense, quantum mechanics makes statistical predictions.
For example, if
$$\int_a^b |\psi(x,t)|^2 dx = 0.30$$
then repeated measurements on identically prepared particles will find the particle between $a$ and $b$ about $30\%$ of the time.
A simple example
Suppose a particle in one dimension has a wave function that is constant between $x=0$ and $x=L$, and zero elsewhere:
$$\psi(x) =
\begin{cases}
A, & 0 \le x \le L \\
0, & \text{otherwise}
\end{cases}$$
Then the probability density is
$$|\psi(x)|^2 =
\begin{cases}
|A|^2, & 0 \le x \le L \\
0, & \text{otherwise}
\end{cases}$$
So the particle is equally likely to be found anywhere between $0$ and $L$. There is no chance of finding it outside that region.
If we want the probability of finding it between $x=L/4$ and $x=L/2$, then
$$P\left(\frac{L}{4} \le x \le \frac{L}{2}\right)
= \int_{L/4}^{L/2} |A|^2 dx
= |A|^2 \left(\frac{L}{2} - \frac{L}{4}\right)
= |A|^2 \frac{L}{4}$$
Once the wave function is normalized, this becomes a definite number.
Normalization idea
Because total probability must be $1$, the wave function must satisfy
$$\int_{-\infty}^{\infty} |\psi(x,t)|^2 dx = 1$$
in one dimension, or
$$\int |\psi(\mathbf{r},t)|^2 dV = 1$$
in three dimensions.
This requirement is called normalization. The detailed method of normalization belongs to its own chapter, but here it is important to see that the probability interpretation only works physically if total probability adds up to $1$.
A physically acceptable wave function must give total probability equal to $1$:
$$\int |\psi|^2 = 1$$
Probability and nodes
Sometimes the wave function is zero at certain points. At those points,
$$|\psi|^2 = 0$$
so the probability of finding the particle there is zero. Such points are often called nodes.
A node does not mean the particle travels around that point like a tiny ball avoiding it in a classical path. It means that if a position measurement is made, that position will never be observed for that quantum state.
Relative probability
Even before full normalization, $|\psi|^2$ can still be used to compare likelihoods. If one region has a larger value of $|\psi|^2$ than another, then the particle is more likely to be found there.
For example, if at one point $x_1$ we have
$$|\psi(x_1)|^2 = 4 |\psi(x_2)|^2$$
then a small interval around $x_1$ is four times as likely to contain the particle as an equally small interval around $x_2$.
This idea is often useful when comparing different parts of a wave pattern.
Visualizing probability density
A graph of $\psi(x)$ and a graph of $|\psi(x)|^2$ are not the same. The wave function may cross zero, change sign, or be complex, while the probability density is always nonnegative.
The table below shows the difference.
| Quantity | Can be negative? | Can be complex? | Direct physical meaning | ||
|---|---|---|---|---|---|
| $\psi(x,t)$ | Yes | Yes | Encodes the quantum state | ||
| $ | \psi(x,t) | ^2$ | No | No | Probability density |
A wave function with large oscillations does not necessarily mean negative probability anywhere. Once we calculate $|\psi|^2$, the result is always a valid density.
A simple sketch
In this sketch, the blue curve for $\psi(x)$ goes above and below zero, while the red curve for $|\psi(x)|^2$ stays nonnegative relative to its own baseline. This illustrates that probability density is obtained from the magnitude squared of the wave function.
Born's rule
The probability interpretation is often called Born's rule, after Max Born. It is one of the central postulates of quantum mechanics. It connects the mathematical object $\psi$ with actual experimental results.
Without this rule, the wave function would be only an abstract mathematical expression. With it, the theory makes testable predictions about measurement outcomes.
Born's rule states that the probability density for finding a particle at position $x$ and time $t$ is
$$|\psi(x,t)|^2$$
What this interpretation does and does not say
The probability interpretation tells us the likelihood of measurement results. It does not say that the particle is spread out like ordinary matter in the classical sense, and it does not say that the particle has a hidden classical path that we simply do not know. At the beginner level, the key point is more modest and more practical: quantum mechanics predicts probabilities, and those probabilities come from $|\psi|^2$.
This interpretation is one of the reasons quantum mechanics looks so different from Newtonian mechanics. In classical physics, we try to predict exact trajectories. In quantum physics, the wave function allows us to predict distributions of outcomes.
Summary
The wave function $\psi$ describes the quantum state, but the measurable probability information comes from $|\psi|^2$. The quantity $|\psi|^2$ is the probability density, and the probability of finding a particle in a region is obtained by integrating this density over that region. Total probability must equal $1$, which leads to normalization. This probabilistic meaning of the wave function is one of the foundations of quantum mechanics.