Determinism of Closed-System Unitary Dynamics

The precise deterministic claim

Let the state of a closed quantum system at time $t$ be represented by a normalized ket

where physically equivalent kets differing only by a global phase represent the same pure state:

The dynamical law in the Schrödinger picture is

where $H(t)$ is the system Hamiltonian. For present purposes, “closed” means that this state and Hamiltonian are intended to describe all dynamically relevant degrees of freedom. A prescribed time-dependent control field may be incorporated into $H(t)$; what matters is that the Hamiltonian is specified rather than being an uncontrolled, unmodelled influence.

The determinism claim has a conditional structure:

Given a complete quantum state at $t_0$ and a specified closed-system Hamiltonian, Schrödinger evolution supplies exactly one state at every later time $t$.

This is not a claim that an experimenter knows the initial state or Hamiltonian exactly. Imperfect state preparation, calibration error, or incomplete environmental modelling can make predictions uncertain. But such uncertainty is epistemic: it concerns our information or modelling. It does not introduce several allowed future states from one exact input state under the model.

This parallels the distinction from the previous lesson. A classical trajectory is deterministic if the complete phase-space point and laws fix its future, even if no observer has enough information to calculate it precisely. Here the role of phase-space point is played by the quantum state.

Schrödinger evolution as an initial-value problem

Choose a fixed orthonormal basis $\{|j⟩\}$, and write

The Schrödinger equation becomes

where

Thus, in a finite-dimensional Hilbert space, Schrödinger evolution is simply a coupled system of first-order linear ordinary differential equations:

For ordinary physical Hamiltonians, the matrix elements are sufficiently regular in time—for example, continuous or piecewise continuous—that standard existence-and-uniqueness results apply. Specifying all components of

therefore selects one solution $\mathbf{\psi }(t)$.

There is an important logical detail here. Uniqueness alone is not enough: a deterministic law needs an evolution to exist as well. In the usual finite-dimensional formulation, existence and uniqueness come together. The formal slogan that “the Schrödinger equation is deterministic” abbreviates both facts.

For infinite-dimensional Hilbert spaces, familiar in wave mechanics and quantum field theory, the statement needs more technical care. One generally requires a suitable self-adjoint Hamiltonian, its domain, and appropriate continuity properties. The mere presence of a formal symbol $H$ in an equation is not itself a proof of a well-defined evolution. But the standard closed-system quantum models used in practice are built precisely to satisfy these requirements.

A direct uniqueness proof

The standard ODE theorem already establishes the result, but it is instructive to see how Hermiticity and linearity rule out two distinct solutions with the same initial state.

Suppose that $|{\psi }_1(t)⟩$ and $|{\psi }_2(t)⟩$ both obey the same Schrödinger equation with the same Hamiltonian and the same initial state:

Define their difference:

Because the Schrödinger equation is linear, the difference also obeys it:

Its initial value is zero:

Now calculate the time derivative of its squared norm:

Using the Schrödinger equation and its adjoint gives

where $H^{†}=H$ has been used. Hence the norm of $|\delta (t)⟩$ is constant. Since it begins at zero, it remains zero:

Therefore,

at all later times. There cannot be two distinct Schrödinger trajectories passing through the same initial ket.

This is exactly the structure of the determinism criterion from the previous lesson:

One small refinement matters physically. Pure states are rays, not individual kets. But evolution respects this equivalence:

So if two initial kets differ only by a global phase, their evolved kets do too. Unitary evolution therefore induces a unique evolution on the physically meaningful space of rays as well.

The time-evolution operator

Rather than solve the differential equation afresh for each initial state, package its solution into a single operator:

The operator $U(t,t_0)$, also called the propagator or time-development operator, is determined by

with initial condition

This formulation makes the deterministic map explicit:

For a time-independent Hamiltonian,

the solution has the familiar exponential form:

For a genuinely time-dependent Hamiltonian, the values $H(t_1)$ and $H(t_2)$ may fail to commute. Then one should not casually replace the solution by an ordinary exponential of an integral. The general expression is a time-ordered exponential:

This complication affects how one calculates the propagator, not whether a specified Hamiltonian defines a unique propagator. Noncommuting Hamiltonians can make the path through time algebraically nontrivial, but they do not add random branching to the state evolution.

The propagators obey the composition rule

and identity rule

These say that evolving in two stages is equivalent to evolving directly over the total interval. They are the quantum analogue of composing deterministic flow maps in classical dynamics.

Why the propagator is unitary

The term unitary evolution means that the propagator preserves inner products:

Equivalently,

To derive this, differentiate $U^{†}U$. From the defining equation for $U$,

Taking the adjoint and using $H^{†}=H$,

Therefore,

Since $U(t_0,t_0)=I$, it follows that

at every time.

Unitarity has several immediate consequences:

• Norms are preserved:

  $$\Vert \psi (t){\Vert }^2=⟨\psi (t)|\psi (t)⟩=⟨\psi (t_0)|\psi (t_0)⟩.$$

  A normalized state remains normalized.

• Transition amplitudes are preserved for two states evolved under the same closed-system dynamics.

• Orthogonal states remain orthogonal. Perfectly distinguishable alternatives do not evolve into indistinguishable ones.

• Evolution is reversible:

  $$U(t,t_0{)}^{-1}=U(t,t_0{)}^{†}=U(t_0,t).$$

It is useful to separate the logical roles:

• Existence and uniqueness establish deterministic evolution.
• Unitarity establishes norm preservation and invertibility.
• In standard closed-system quantum mechanics, both properties arise from the same Schrödinger dynamics with a self-adjoint Hamiltonian.

Thus, unitarity is stronger than mere forward determinism. A noninvertible map can still have one unique future for every state, as we saw in the previous lesson’s discrete examples. But closed-system quantum evolution is not merely forward deterministic: it is a one-to-one, reversible flow on quantum state space.

Worked example: deterministic phase evolution of a qubit

Take a qubit with Hamiltonian

where

For an initial state

the time-evolution operator is

Because $|0⟩$ and $|1⟩$ are energy eigenstates,

and

Linearity then gives the unique evolved state:

Factoring out the physically irrelevant global phase $e^{-i\omega t/2}$, the same ray may be written as

The state’s relative phase rotates at a rate fixed by $\omega$. On the Bloch sphere, this is a rotation about the $z$-axis. Nothing in the state evolution chooses randomly between alternatives: once $\alpha$, $\beta$, and $H$ are specified, the quantum state at every later time is fixed.

The same conclusion holds for a mixed initial state. If the closed system is described by a density operator $\rho (t_0)$, then

This is again a unique map from the initial state to the later state. A density operator may encode classical preparation uncertainty or entanglement with degrees of freedom included in the description, but its closed-system evolution remains unitary and deterministic.

What this result does—and does not—show

The result is powerful, but it has a carefully limited target.

It does establish that textbook quantum mechanics contains a deterministic dynamical law for closed-system states:

It does not establish all of the following:

1. That the quantum state is ontologically complete.
   A deterministic hidden-variable theory could propose a larger state space. Whether such proposals can satisfy locality, noncontextuality, or other principles is a later question.

2. That an observer can predict every practical event.
   Exact state and Hamiltonian knowledge may be unavailable, and complex unitary dynamics can be computationally intractable.

3. That every measurement has one pre-fixed outcome.
   This is the crucial unresolved issue at this stage. Schrödinger evolution uniquely fixes the joint state of a system and apparatus, but it does not by itself explain why an observer records one definite result rather than an entangled superposition of result records.

That final distinction is the bridge to the next lesson. Saying “quantum mechanics is indeterministic” without specifying whether one means state evolution or measurement outcomes conflates two different claims.

Key takeaways

• A closed quantum system obeys

  $$i\hslash \frac{d}{dt}|\psi (t)⟩=H(t)|\psi (t)⟩.$$

• In a fixed basis, this is a system of coupled first-order linear differential equations.

• Under the usual regularity conditions, an initial quantum state and specified Hamiltonian yield an existing, unique solution.

• The resulting propagator,

  $$|\psi (t)⟩=U(t,t_0)|\psi (t_0)⟩,$$

  explicitly represents deterministic state evolution.

• Hermiticity of the Hamiltonian makes the propagator unitary:

  $$U^{†}U=I.$$

  Unitary evolution preserves normalization, inner products, distinguishability, and permits inverse evolution.

• Deterministic unitary evolution of the quantum state is not yet a deterministic account of individual measurement outcomes.

Next, we will examine that gap directly: how a unique unitary evolution can produce an entangled system–apparatus state without, by itself, selecting one unique observed outcome.