Complex Conjugate

For any complex number $z = a + bi$ , the complex conjugate is

$z^* = a - bi.$

Geometrically, conjugation reflects the point across the real axis in the complex plane: the real part $a$ is unchanged and the imaginary part changes sign.

Why conjugates matter for quantum mechanics

The central reason to care about $z^*$ in quantum mechanics is the modulus formula. Multiply $z$ by its own conjugate:

$z \, z^* = (a + bi)(a - bi) = a^2 + b^2.$

The cross-terms $abi$ and $-abi$ cancel, leaving a real non-negative number. Taking the square root gives the modulus:

$|z| = \sqrt{z \, z^*} = \sqrt{a^2 + b^2}.$

In the Born rule, a quantum state assigns a complex amplitude $\alpha$ to each outcome; the probability of that outcome is $|\alpha|^2 = \alpha \, \alpha^*$ . Every probability you will ever compute in quantum mechanics goes through this product.

Key properties

1. Double conjugation is the identity. $(z^*)^* = z$ . Conjugating twice recovers the original number.

2. Conjugation distributes over addition and multiplication. $(z_1 + z_2)^* = z_1^* + z_2^*$ and $(z_1 z_2)^* = z_1^* \, z_2^*$ . These identities are used constantly when simplifying inner products of quantum states.

3. Real numbers are their own conjugates. If $b = 0$ then $z^* = a = z$ . In particular, $|z|^2 = z \, z^*$ is always real and non-negative, as a probability must be.

A worked example

Let $z = 3 - 5i$ . Then $z^* = 3 + 5i$ and

$z \, z^* = (3 - 5i)(3 + 5i) = 9 - 25i^2 = 9 + 25 = 34,$

so $|z| = \sqrt{34} \approx 5.831$ .

Note the step $-25i^2 = 25$ because $i^2 = -1$ . Watch for sign errors here — they are by far the most common mistake when computing moduli.

Try it

This is a numerical exercise — your code should return a number. Use the conjugate identity $|z|^2 = re^2 + im^2$ to find the modulus of $z = 5 - 12i$ (a 5–12–13 right triangle).

Run your code to see the quantum state.