The state evolves. What kind of mathematical object describes this transformation?
Operator view
U·|ψ⟩ = |ψ(t)⟩
→
Matrix × vector
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→
Result
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Properties verified
‖Uψ‖=1U†U=Ireversiblelinear
Timet = 0
0π/2π3π/22π
Setup
State
Properties
Discover
Initial State
Rotation Axis
U(t) — Current evolution operator
Move the slider to see U(t).
the question
You already know the state evolves continuously. Now ask: what mathematical object transforms |ψ(0)⟩ into |ψ(t)⟩? Move the slider and watch the matrix-vector multiplication update in real time.
Before — |ψ(0)⟩
|ψ⟩ = 1|0⟩
↓ U(t)
After — |ψ(t)⟩
|ψ⟩ = 1|0⟩
Probabilities
|0⟩
100%
|1⟩
0%
‖|ψ(t)⟩‖² =1.0000 ✓
before and after
The operator U transforms the state. Compare |ψ(0)⟩ and |ψ(t)⟩. The probabilities change — but the total always sums to 1. The operator preserves something fundamental.
These are evolutions at specific angles — each is a unitary operator with a name. The Bloch sphere on the left animates as you select each one.
Matrix
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What it does to |ψ(0)⟩
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Norm preservation unverified
The total probability must always be 1. If evolution destroyed or created probability, it would destroy information about the quantum state.
‖U|ψ⟩‖ = ‖|ψ⟩‖ = 1
Unitary condition unverified
For U to preserve the norm for every possible state, it must satisfy U†U = I. This is not a choice — it is forced by the physics.
U†U = I
Invertibility unverified
You already saw evolution is reversible — drag the slider backward. This means U has an inverse. For unitary operators, the inverse is U†.
U⁻¹ = U†
Linearity unverified
Evolution is linear: if you superpose two states and then evolve, you get the same result as evolving each state separately and then superposing.
U(α|ψ⟩+β|φ⟩) = αU|ψ⟩ + βU|φ⟩
Inner-product preservation consequence
Because U is unitary, it preserves the overlap (inner product) between any two states. Transition probabilities are unchanged by the evolution frame.
⟨Uψ|Uφ⟩ = ⟨ψ|φ⟩
Hermitian conjugate U† tool
To compute U†: transpose the matrix, then complex-conjugate every entry. It appears throughout: U†U = I, and U⁻¹ = U†.
(U†)ᵢⱼ = (Uⱼᵢ)*
Eigenvalues optional
Every eigenvalue of a unitary operator has magnitude 1. They live on the unit circle in the complex plane — pure phase factors e^(iθ).
|λ| = 1 → λ = eiθ
the key insight
These properties are not arbitrary math rules. Each one follows directly from a physical requirement: probability is conserved, information is not lost, evolution can be undone.
You are about to evolve a qubit with operator U. Predict: will ‖|ψ(t)⟩‖ — the total probability — change as t increases?
‖|ψ(t)⟩‖ stayed exactly 1.0000. Not approximately — exactly. This is not a coincidence. The operator U is norm-preserving by construction. Any operator that fails this test cannot describe valid quantum evolution.
The matrix U evolves the state. What is U†U — the Hermitian conjugate multiplied by U itself?
U†U = I always — regardless of t, regardless of the initial state. This is the unitary condition. It guarantees norm preservation for every possible quantum state, not just the one you tried.
If U evolves |ψ(0)⟩ forward to |ψ(t)⟩, which operator reverses the evolution — returning the state to |ψ(0)⟩?
U† reverses the evolution exactly. This is what U⁻¹ = U† means — the inverse of a unitary operator is its Hermitian conjugate. Evolution is always reversible; information is always preserved.
★ Final Challenge
A new operator V satisfies V†V = I. You apply it to state |+⟩. Without running the simulation: what is ‖V|+⟩‖?
‖V|+⟩‖ = exactly 1. The condition V†V = I alone guarantees this — for any state, not just |+⟩. You applied the mathematical property directly, without needing to know what V is. That is what it means to understand the operator, not just memorize it.
Describing Evolution · F2
Investigation complete.
You derived why evolution must be unitary — not from a textbook definition, but from the physical requirements of probability conservation, reversibility, and linearity.