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Evolution

Abstract

The details of how quantum states evolve.

Keywords:UnitariesDynamicsEvolution.

Time Evolution

The time evolution of a closed quantum system (one not interacting with the environment and hence described by a ket vector) is given by the Schrödinger equation,

iℏddt∣ψ⟩=H∣ψ⟩,i \hbar \frac{d}{dt} \ket{\psi} = H \ket{\psi},
(1)

where HH is a hermitian operators called the Hamiltonian. The Hamiltonian describes the energy landscape the system exists within and hence is dependent on the system being modelled.

If a state ∣ψ⟩\ket{\psi} is evolved into a state ∣ψ′⟩\ket{\psi'} from t1t_1 to t2t_2, the solution to the Schrödinger equation is

∣ψ′⟩=exp⁡[−iH(t1−t2)ℏ]∣ψ⟩.\ket{\psi'} = \exp \bigg[ \frac{-iH(t_1 - t_2)}{\hbar} \bigg] \ket{\psi}.
(2)

This is a unitary operator, meaning that the time evolution of a closed quantum system is given by the action of a unitary operator on the current state.

Proof

The following facts will be used

  1. The exponential of an operator XX is given by

eX=∑k=0∞Xkk!=I+X+X22+X36+…e^{X} = \sum_{k=0}^{\infty} \frac{X^k}{k!} = \mathbb{I} + X + \frac{X^2}{2} + \frac{X^3}{6} + \ldots
(3)

from this one gets

  1. eAeB=eA+Be^{A}e^{B}=e^{A+B} if [A,B]=0[A,B] = 0

  2. (eA)†=eA†(e^{A})^{\dagger} = e^{A^{\dagger}} as (An)†=(AA…A)†=(A†)n(A^{n})^{\dagger} = (AA \ldots A)^{\dagger} = (A^{\dagger})^{n}

Let

U=eAU = e^{A}
(4)

where

A=−iH(t1−t2)ℏ.A = \frac{-iH(t_1 - t_2)}{\hbar}.
(5)

Hence,

A†=[−iH(t1−t2)ℏ]†=iH†(t1−t2)ℏ=iH(t1−t2)ℏ=−A,\begin{align*} A^{\dagger} &= \bigg[ \frac{-iH(t_1 - t_2)}{\hbar} \bigg]^{\dagger} \\ &= \frac{iH^{\dagger}(t_1 - t_2)}{\hbar} \\ &= \frac{iH(t_1 - t_2)}{\hbar} \\ &= -A, \end{align*}
(6)

where the fact that H=H†H=H^{\dagger} has been used. This means that

U†=[eA]†=eA†=e−A.\begin{align*} U^{\dagger} &= \big[ e^{A} \big]^{\dagger} \\ &= e^{A^{\dagger}} \\ &= e^{-A}. \end{align*}
(7)

Hence,

UU†=eAe−A,=eA−A=e0,\begin{align*} UU^{\dagger} &= e^{A}e^{-A}, \\ &= e^{A-A} \\ &= e^{0}, \end{align*}
(8)

where 0 is the zero operator. The operator expansion of the exponential of the zero operator then gives the identity,

e0=I.e^{0} = \mathbb{I}.
(9)

Repeating for U†UU^{\dagger}U completes the proof.

One can therefore succinctly model the evolution of closed systems in quantum theory through unitary operators,

∣ψ′⟩=U∣ψ⟩,  UU†=U†U=I,\ket{\psi'} = U \ket{\psi}, ~ ~ UU^{\dagger} = U^{\dagger}U = \mathbb{I},
(10)

without concern for what the specific Hamiltonian is.

Properties of Unitary Operators

Unitary operators have the following properties:

  1. They act linearly on superpositions

    • Let ∣+⟩=12(∣0⟩+∣1⟩)∈H\ket{+} = \frac{1}{\sqrt{2}} \big( \ket{0} + \ket{1}) \in \mathcal{H}, then

U∣+⟩=12(U∣0⟩+U∣1⟩)U \ket{+} = \frac{1}{\sqrt{2}} \big( U\ket{0} + U\ket{1})
(11)

  1. They take normalised states to normalised states

    • Let ∣ψ⟩∈H\ket{\psi} \in \mathcal{H} such that ⟨ψ∣ψ⟩=1 \braket{\psi|\psi} = 1 and let ∣ψ′⟩=U∣ψ⟩\ket{\psi'} = U \ket{\psi}, then

⟨ψ′∣ψ′⟩=⟨ψ∣U†U∣ψ⟩=⟨ψ∣ψ⟩=1,  as  U†U=UU†=I.\braket{\psi'|\psi'} = \bra{\psi}U^{\dagger}U\ket{\psi} = \braket{\psi|\psi}=1,~~{\rm as}~ ~ U^\dagger U=UU^\dagger =\mathbb{I}.
(12)

  1. They represent reversible dynamics.

    • If UU is a unitary operator, then V=U†V=U^\dagger is a unitary operator. Hence, if there exists a unitary UU evolving a state ∣ψ⟩→∣ψ′⟩\ket{\psi} \rightarrow \ket{\psi'}, then there always exists a unitary operators VV evolving ∣ψ′⟩→∣ψ⟩\ket{\psi'} \rightarrow \ket{\psi}. In practice, reversing a unitary evolution, or even finding VV from UU can be very difficult.

Quantum Formalism
Quantum States
Quantum Formalism
Observables