Superposition

Superposition is the idea that a valid quantum state can be a linear combination of basis states, not only $\lvert 0 \rangle$ or $\lvert 1 \rangle$ alone.

Key idea

Prepare states like $\lvert \psi \rangle = \alpha \lvert 0 \rangle + \beta \lvert 1 \rangle$ with complex amplitudes satisfying $\lvert \alpha \rvert^2 + \lvert \beta \rvert^2 = 1$. The values $\lvert \alpha \rvert^2$ and $\lvert \beta \rvert^2$ are outcome probabilities for standard measurement in the computational basis.

A familiar analogy: waves add up

In everyday life, you have already seen superposition—just not the quantum kind.

• Sound: when you strum a guitar chord, waves from different strings overlap. At each point in space and time, the total pressure fluctuation is the sum of contributions from each string.
• Water: ripples from multiple pebbles thrown into a pond overlap and mix as they travel outward. Where crests meet crests they grow; where crests meet troughs they cancel.

This is classical wave superposition: a point-by-point addition that produces a new wave pattern.

What makes quantum superposition different?

Quantum systems can also be described by waves, but the wave is not “a wave of stuff like water.” Instead, a quantum state encodes probability amplitudes.

Roughly speaking:

• the wave-like pattern (amplitudes and phases) evolves smoothly,
• measurement produces a discrete outcome,
• the pattern determines the probabilities of those outcomes.

Example: electrons in atoms (orbitals)

An electron bound to an atom is not located at a single, definite point like a planet. Instead, it is described by an orbital—a 3D wave-like distribution that encodes where the electron is likely to be found if you measure its position.

Different energy levels correspond to different orbital shapes. You can also prepare a superposition of energy eigenstates, schematically:

The resulting probability pattern changes over time and can affect measurable properties of the system.

The double-slit experiment (why it feels weird)

In the standard double-slit setup, electrons are fired one at a time toward two narrow slits, and a detector screen records where they land.

If electrons behaved like tiny classical particles, you would expect two clusters—one behind each slit. Instead, when no which-path information is available, the accumulated hits form an interference pattern, consistent with a wave passing through both slits and interfering with itself.

One useful way to phrase this is:

• the electron is in a superposition of “went through the left slit” and “went through the right slit,”
• the corresponding probability amplitudes interfere,
• measurement at the screen yields a single dot, but many dots reveal the interference structure.

Common confusion (and a better mental model)

It is common to hear that a quantum system is “in many places at once.” This is a shorthand that often causes confusion.

A safer mental model is:

• the quantum state is a normalized vector (a wave-like object),
• it can have support across space (or across multiple basis states),
• and it encodes measurement statistics plus phase relationships that enable interference.

A hands-on demonstration: polarization filters

Light provides one of the cleanest ways to see superposition-like behavior in the lab. Many light sources emit a mix of polarizations, and polarizing filters project that light onto a chosen axis.

The “three polarizers” surprise

Consider three polarizers:

• the first passes horizontal polarization (H),
• the last passes vertical polarization (V),
• the middle is set to 45° (diagonal, D).

If you place H followed by V, essentially no light gets through. That makes sense: after H, the light is horizontally polarized; V blocks horizontal light.

But if you insert a diagonal polarizer between them (H → D → V), some light does pass through. The middle filter changes the basis and “re-prepares” the polarization component that has a nonzero projection onto V.

A simple quantitative explanation (Malus’ law)

For ideal polarizers, the transmitted intensity follows Malus’ law:

• For H → V, $\Delta \theta = 90^\circ$, so $I = I_0 \cos^2(90^\circ) = 0$.
• For H → D, $\Delta \theta = 45^\circ$, so $I = I_0 \cos^2(45^\circ) = \frac{I_0}{2}$.
• For D → V, again $45^\circ$, so $I = \frac{I_0}{2} \cos^2(45^\circ) = \frac{I_0}{4}$.

So the three-filter setup transmits about 25% of the original intensity (in the idealized case).

Next

Continue to Entanglement.

Next

Continue to Entanglement.