Time-Evolution Operator: Derivation and Unitarity

Introduction

Welcome to the next lesson in our journey through the foundations of quantum mechanics. In our previous session, we established the generalized uncertainty principle, a direct consequence of the Hilbert space structure that quantifies the fundamental trade-off in measuring non-commuting observables. This principle governs the static properties of a quantum state at a fixed moment in time. Our focus now shifts from the static to the dynamic: how do quantum states evolve?

Today, we will answer this question by formally deriving the operator that governs time evolution. Starting from the fundamental physical requirement of probability conservation, we will prove that this operator must be unitary. We will then see how this property, when applied to an infinitesimal time step, directly yields the time-dependent Schrödinger equation. This lesson will establish the crucial link between the Hamiltonian as the generator of time evolution and the unitary operator that enacts this evolution over finite time intervals.

The Time-Evolution Operator and Unitarity

The fourth postulate of quantum mechanics asserts that the evolution of a quantum state $∣ ψ (t_{0})⟩$ at an initial time $t_{0}$ to a state $∣ ψ (t)⟩$ at a later time $t$ is described by a linear operator, $U (t, t_{0})$ , known as the time-evolution operator:

∣ ψ (t)⟩ = U (t, t_{0}) ∣ ψ (t_{0})⟩

Before we determine the form of this operator, we can deduce its most critical property from a single physical principle: the total probability of finding the particle somewhere must remain 1 at all times. This conservation of probability means the norm of the state vector must be constant.

The following video provides a lucid and rigorous argument, starting from basic physical intuition, to prove that this requirement forces the time-evolution operator to be unitary.

Ch 13: Where does the Schrödinger equation come from? | Maths of Quantum Mechanics

This segment from the "Quantum Sense" channel, titled "Where does the Schrödinger equation come from?", builds the case for unitarity from the ground up. It's an excellent example of how physical principles constrain the mathematical formalism of the theory.

Please watch the segment from the introduction of the operator to the conclusion of the unitarity proof. Pay close attention to how two key ideas—the reversibility of time evolution and the conservation of probability—are used to prove that U^\dagger U = \mathbb{1}.

As you just saw, the condition $⟨ ψ (t) ∣ ψ (t)⟩ = ⟨ ψ (t_{0}) ∣ ψ (t_{0})⟩$ for any state $∣ ψ ⟩$ is mathematically equivalent to the statement that $U (t, t_{0})$ is a unitary operator, satisfying $U^{†} (t, t_{0}) U (t, t_{0}) = 1$ . This is a profound connection: the conservation of probability, a cornerstone of any consistent physical theory, is encoded in the unitarity of the time-evolution operator.

From Unitarity to the Schrödinger Equation

Now, let's see how the Schrödinger equation emerges directly from this property of unitarity. The key is to consider an infinitesimal time evolution from $t$ to $t + d t$ . We can expand the operator $U (t + d t, t)$ in a Taylor series around $d t = 0$ :

U (t + d t, t) = U (t, t) + \frac{d U}{d t ^{'}}_{t^{'} = t} d t + O (d t^{2})

Since $U (t, t) = 1$ , and we can define an operator $G^(t)$ as the generator of the evolution, we can write:

U (t + d t, t) \approx 1 - \frac{i}{ℏ} G^(t) d t

The factor of $- i /ℏ$ is a convention at this stage, but its convenience will soon become apparent. Now, let's impose the condition that this infinitesimal operator is unitary:

U^{†} (t + d t, t) U (t + d t, t) = 1

(1 + \frac{i}{ℏ} G^^{†} (t) d t) (1 - \frac{i}{ℏ} G^(t) d t) = 1

Expanding this product and keeping terms only to the first order in $d t$ , we find:

1 + \frac{i}{ℏ} (G^^{†} (t) - G^(t)) d t = 1

This implies that $G^^{†} (t) - G^(t) = 0$ , which means the generator $G^(t)$ must be a Hermitian operator. This generator of time translations is identified as the system's Hamiltonian, $H^(t)$ .

The statement that the generator of time evolution must be Hermitian is equivalent to the statement that time evolution is unitary.

With this, we can now derive the differential equation for the state vector $∣ ψ (t)⟩$ :

∣ ψ (t + d t)⟩ = U (t + d t, t) ∣ ψ (t)⟩ \approx (1 - \frac{i}{ℏ} H^(t) d t) ∣ ψ (t)⟩

Rearranging and taking the limit as $d t \to 0$ :

d t \to 0 lim \frac{∣ ψ ( t + d t )⟩ - ∣ ψ ( t )⟩}{d t} = \frac{d}{d t} ∣ ψ (t)⟩ = - \frac{i}{ℏ} H^(t) ∣ ψ (t)⟩

Multiplying by $i ℏ$ gives the familiar time-dependent Schrödinger equation:

i ℏ \frac{d}{d t} ∣ ψ (t)⟩ = H^(t) ∣ ψ (t)⟩

The following resource provides a written account of this derivation, cementing the link between the unitarity of $U$ and the Hermitian nature of $H$ .

[PDF] Quantum Theory I, Lecture 6 Notes - MIT OpenCourseWare

These lecture notes from the MIT course on Quantum Theory I formalize the arguments we've just discussed.

Please read section 6.2, starting from the heading "Quantum Dynamics" and continuing through the derivation that shows H must be Hermitian. Then, read the short passage that follows, which shows how this leads directly to the Schrödinger equation.

The Formal Solution for the Time-Evolution Operator

Having derived the Schrödinger equation from the properties of $U$ , we can now solve for $U$ itself. By substituting $∣ ψ (t)⟩ = U (t, t_{0}) ∣ ψ (t_{0})⟩$ into the Schrödinger equation, we obtain a differential equation for the operator $U$ :

i ℏ \frac{d}{d t} U (t, t_{0}) = H^(t) U (t, t_{0})

with the initial condition $U (t_{0}, t_{0}) = 1$ .

Case 1: Time-Independent Hamiltonian

Let's first consider the simplest case, where the Hamiltonian $H^$ does not depend on time. The operator differential equation is directly analogous to the scalar equation $y^{'} (t) = k y (t)$ , whose solution is $y (t) = e^{k t} y (0)$ . The solution for the operator is thus a matrix exponential:

U (t, t_{0}) = exp (- \frac{i H ^ ( t - t _{0} )}{ℏ})

where the exponential of an operator $A^$ is defined by its Taylor series expansion, $e^{A^} = \sum_{n = 0}^{\infty} \frac{A ^ ^{n}}{n !}$ . Since $H^$ is Hermitian, it is straightforward to show that this form of $U$ is indeed unitary.

Case 2: Time-Dependent Hamiltonian (A Deeper Look)

When $H^$ depends on time, the solution is more complex. The naive solution, $exp (- \frac{i}{ℏ} \int_{t_{0}}^{t} H^(t^{'}) d t^{'})$ , is incorrect because the Hamiltonian at different times, $H^(t_{1})$ and $H^(t_{2})$ , may not commute. The non-commutativity prevents the simple combination of terms in the Taylor series expansion.

The correct solution requires ordering the operators in the expansion chronologically. This leads to the Dyson series, which can be written compactly using the time-ordering operator $T$ :

U (t, t_{0}) = T exp (- \frac{i}{ℏ} \int_{t_{0}}^{t} H^(t^{'}) d t^{'})

The time-ordering operator $T$ arranges any product of time-dependent operators so that their time arguments increase from right to left. For example, $T [A^(t_{1}) B^(t_{2})] = A^(t_{1}) B^(t_{2})$ if $t_{1} > t_{2}$ , but $T [A^(t_{1}) B^(t_{2})] = B^(t_{2}) A^(t_{1})$ if $t_{2} > t_{1}$ .

Given your current studies in QFT, this formalism should be particularly relevant, as the Dyson series is the cornerstone of time-dependent perturbation theory and the calculation of S-matrix elements. The following lecture provides a superb, graduate-level derivation of both the time-independent solution and the Dyson series.

Quantum Theory, Lecture 4.5: Time Evolution Operator. Dyson Series. Time-Ordered Exponential.

This lecture by Prof. Alexander Maloney offers a rigorous treatment of the time-evolution operator, perfectly suited to your background. It moves from the foundational concepts to the advanced formalism of the Dyson series.

Please watch the following segments: Operator and Unitarity: Watch from this point to see the introduction of the time-evolution operator U and a rigorous proof of its unitarity and the consequences for its generator, the Hamiltonian. This provides an excellent alternative perspective to the first video. Time-Independent Solution: The segment from here explicitly solves the operator differential equation for a time-independent Hamiltonian, yielding the familiar exponential form. Time-Dependent Case and Dyson Series: This is the most advanced part. Watch from this section. Prof. Maloney masterfully explains why the simple exponential integral fails when Hamiltonians at different times do not commute and then carefully constructs the time-ordered exponential (Dyson series) as the correct solution. Focus on the conceptual leap from the time-independent to the time-dependent case: the problem is no longer just solving a DE, but managing the non-commutativity of operators across time.

Conclusion

In this lesson, we have laid the dynamical foundation of quantum mechanics. We established that the evolution of a quantum state is governed by a unitary operator, a property that is mathematically equivalent to the conservation of total probability.

Key Takeaways:

The evolution of a state $∣ ψ (t_{0})⟩$ to $∣ ψ (t)⟩$ is described by the unitary time-evolution operator $U (t, t_{0})$ .
Unitarity ( $U^{†} U = 1$ ) is a direct and necessary consequence of probability conservation.
The generator of infinitesimal time evolution is the Hermitian Hamiltonian operator, $H^$ . The relationship between the two leads directly to the time-dependent Schrödinger equation: $i ℏ \frac{d}{d t} ∣ ψ ⟩ = H^∣ ψ ⟩$ .
For a time-independent Hamiltonian, the evolution operator has the form $U (t, t_{0}) = e^{- i H^(t - t_{0}) /ℏ}$ .
For a time-dependent Hamiltonian, the solution is the time-ordered exponential, or Dyson series, which correctly handles the non-commutativity of the Hamiltonian at different times.

We have seen that time evolution resides in the state vectors. But is this the only way to view quantum dynamics? In our next lesson, we will use the time-evolution operator to explore different but equivalent formulations of quantum mechanics—the Schrödinger, Heisenberg, and Interaction pictures—where the responsibility for time evolution can be shifted from the states to the operators.

Recommended resources