The Wavefunction and Probability

Manual: General · Subject: Physics

Learn how quantum physics uses the wavefunction to predict probabilities.

The language of quantum mechanics

What is the wavefunction?

The wavefunction, usually written as $\psi$ , is a mathematical object that contains all the information we can know about a quantum system. It does not directly give a measurement result, but it lets us calculate probabilities.

Probability from amplitude

The square of the wavefunction’s magnitude, written as $|\psi|^2$ , gives the probability density of finding a particle in a certain place or state. This idea is often called the Born rule.

Normalization

Because total probability must equal $1$ , the wavefunction must be normalized. In simple form, this means $\int |\psi|^2\,dx = 1$ over all possible positions.

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Main idea

Quantum theory predicts probabilities, not exact outcomes for individual measurements.

Why this is different

In classical physics, if you know the initial conditions exactly, you can often predict the future exactly. In quantum physics, even complete knowledge of the wavefunction only gives probabilities for measurement results.

What does $|\psi|^2$ usually represent?

The exact position of the particle The probability density The mass of the particle The particle's temperature

What is the symbol commonly used for the wavefunction?

Reading a wavefunction

1

Step 1: Write down the wavefunction $\psi$ .
2

Step 2: Compute $|\psi|^2$ .
3

Step 3: Interpret $|\psi|^2$ as probability density.
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Step 4: Normalize so total probability is $1$ .

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