Superposition & the State — Knowledge Base

The state vector

In classical physics, the state of a system is a list of numbers: position, momentum, energy — each with a definite value. In QM, the state is a vector — a list of amplitudes, one for each possible outcome of a measurement.

If a measurement has N possible outcomes, the quantum state is a vector with N entries. Each entry is a probability amplitude (a complex number). Its absolute square is the probability of getting that outcome if you measure now.

A quantum state is a list of probability amplitudes, one per possible measurement outcome. The amplitude squared gives the probability. The state is normalised — probabilities must sum to 1.

Amplitude for up — squared gives P(↑)
Amplitude for down — squared gives P(↓)
State vector — the full description of the system

The simplest non-trivial quantum system has exactly two possible outcomes. This is called a two-level system, and the physical example is the spin of an electron — spin-½. It can be measured as "up" or "down" along any axis. Before measurement, it can be in any superposition of both.

The state is not a probability distribution. A probability distribution says "I am ignorant." A quantum superposition says "the system has both possibilities simultaneously, and will behave as if it explores both until forced to pick one." The difference is the interference pattern on the double-slit screen.

Bases and representation

"Up" and "down" along the z-axis are one possible basis — one way of asking a question. But spin-½ has infinitely many bases. You can measure along x, y, z, or any axis. The state vector is the same physical object; what changes is the basis you write it in.

This is exactly like how a 2D vector can be described in Cartesian coordinates (x, y) or polar coordinates (r, θ). It is the same arrow. The coordinates depend on your choice of axes. The arrow does not.

Both diagrams show the same quantum state |ψ⟩. In the z-basis (left), its coordinates are (a↑, a↓). In the x-basis (right), its coordinates are (b→, b←). The state is the same; the basis is a choice of which measurement to perform.

State vector — the same physical object in both diagrams
Basis components — coordinates depend on the measurement axis
Basis label — the measurement you are asking about

Choosing a basis is choosing a question. "Is the spin up or down?" (z-basis) and "Is the spin left or right?" (x-basis) are different questions. The state has amplitudes for all of them. The answer you get depends on which question you ask. Neither is more "real" — they are complementary descriptions of the same state.

The measurement postulate

In QM, measurement is not a passive observation. It is an active process with two distinct steps:

Step 1: Collapse
The state vector jumps to one of its basis vectors — the one corresponding to the outcome. All other possibilities vanish instantly. This is not a physical process. It is the update of what you can predict given the new information.

Step 2: Outcome
The probability of each outcome is the square of its amplitude. The outcome is fundamentally random — there is no hidden variable that determines it in advance. The randomness is irreducible.

Before measurement, the state is a superposition. When measured, it jumps (collapses) to one of the basis states. The probability of each outcome is the squared amplitude. The randomness is irreducible — no hidden variable determines which outcome occurs.

Before measurement — all possibilities simultaneously
After measurement — exactly one outcome, random
The act of measurement — collapses the wavefunction

The measurement postulate is not a theory of measurement. It is a rule that correctly predicts the outcomes of experiments. Why collapse happens — whether it is a physical process, a Bayesian update, or branching worlds — is not settled physics. What is settled is that the rule works, and no alternative rule makes different predictions that are also correct.

Spin in a magnetic field: Stern-Gerlach

The simplest physical realisation of the two-level system is an electron (or a silver atom) passing through an inhomogeneous magnetic field. The field splits the beam into two — spin up and spin down — by a tiny amount. That is the Stern-Gerlach experiment.

But here is the quantum part: if you take the "up" output and send it through another Stern-Gerlach device oriented sideways, it splits into "left" and "right" with equal probability. The spin that was definitely up (in the z-basis) is in a superposition of left and right (in the x-basis). The state was perfectly known in one basis and completely uncertain in another.

_z → equal superposition of →_x and ←_x Certainty in one basis means maximum uncertainty in a complementary basis.

A spin that is definitely up (z-basis) enters a sideways Stern-Gerlach and splits 50/50 into left and right (x-basis). Knowing the answer to one question completely destroys the answer to another.

Spin up (z-basis) — definite answer
Spin down (z-basis) — the other definite answer
Spin left/right (x-basis) — 50/50 superposition of a definite z-state

The Stern-Gerlach experiment makes one thing inescapably clear: the quantum state is not a thing with a hidden pre-existing value. If spin up (z) were really "spin up with a hidden value for x," then 100% of the up beam would go one way. It does not. The x-spin does not exist until you measure it.

What the state vector gives us

Concept	Classical equivalent	Quantum reality
State	List of definite properties	Vector of amplitudes for all possible measurement outcomes
Basis	Choosing coordinate axes — arbitrary, no consequence	Choosing what to measure — determines which outcomes are possible
Measurement	Reading a pre-existing value	Collapsing the state onto a basis vector — creates the outcome
Complementarity	Does not exist — you can measure position and momentum simultaneously	Some bases are incompatible — measuring in one destroys information about another
Uncertainty	Ignorance — could be reduced with more data	Irreducible — the state does not have a value for that basis until measured

The state vector is the complete description of a quantum system at a moment in time. It evolves according to the Schrödinger equation (a wave equation — deterministic, smooth, reversible) except when measured. At measurement, it jumps — irreversibly, randomly — to one of the basis vectors. Two rules, one for each regime: smooth evolution and sudden collapse.

The deepest open question: Why these two rules? Why does a quantum system follow a smooth wave equation most of the time and then, on measurement, jump discontinuously? This is the measurement problem. Every interpretation of QM — many-worlds, Bohmian mechanics, QBism, objective collapse — is an attempt to answer this. None has settled it experimentally. The rules themselves are not in doubt.

Grok check

Prediction, not recall.

A spin-½ system is in the state |ψ⟩ = 0.6|↑⟩ + 0.8|↓⟩. What is the probability of measuring spin up? Spin down? Do these sum to 1?
You measure spin up (z-basis). The state collapses to |↑⟩. You immediately measure again in the same basis. What do you get? What if you measure in a different basis first?
A state has equal amplitudes for |↑⟩ and |↓⟩ along z: |ψ⟩ = 1/√2 (|↑⟩ + |↓⟩). What is the probability of each? If you measure along x instead, what do you expect? (Hint: what did the Stern-Gerlach experiment show?)
The state vector is "the complete description" of a quantum system. Yet it can only predict probabilities, not certainties. In what sense is a probability distribution a "complete" description?

Question 4 is the most philosophically charged. The answer is that completeness does not mean certainty — it means nothing is missing. No hidden variable, no extra data, no deeper level of reality. The probability is the final answer. That is what "irreducible randomness" means, and it is what makes QM fundamentally different from every earlier theory.

Next: Uncertainty & Complementarity — why you cannot know everything, and why quantum objects seem contradictory.