Abstract¶ The details of how quantum states evolve.
Keywords: Unitaries Dynamics Evolution. ¶ 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,
i ℏ d d t ∣ ψ ⟩ = H ∣ ψ ⟩ , i \hbar \frac{d}{dt} \ket{\psi} = H \ket{\psi}, i ℏ d t d ∣ ψ ⟩ = H ∣ ψ ⟩ , where H H H 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} ∣ ψ ⟩ ∣ ψ ′ ⟩ \ket{\psi'} ∣ ψ ′ ⟩ t 1 t_1 t 1 t 2 t_2 t 2
∣ ψ ′ ⟩ = exp [ − i H ( t 1 − t 2 ) ℏ ] ∣ ψ ⟩ . \ket{\psi'} = \exp \bigg[ \frac{-iH(t_1 - t_2)}{\hbar} \bigg] \ket{\psi}. ∣ ψ ′ ⟩ = exp [ ℏ − i H ( t 1 − t 2 ) ] ∣ ψ ⟩ . This is a unitary operator
The following facts will be used
The exponential of an operator X X X e X = ∑ k = 0 ∞ X k k ! = I + X + X 2 2 + X 3 6 + … e^{X} = \sum_{k=0}^{\infty} \frac{X^k}{k!} = \mathbb{I} + X + \frac{X^2}{2} + \frac{X^3}{6} + \ldots e X = k = 0 ∑ ∞ k ! X k = I + X + 2 X 2 + 6 X 3 + … from this one gets
e A e B = e A + B e^{A}e^{B}=e^{A+B} e A e B = e A + B [ A , B ] = 0 [A,B] = 0 [ A , B ] = 0 ( e A ) † = e A † (e^{A})^{\dagger} = e^{A^{\dagger}} ( e A ) † = e A † ( A n ) † = ( A A … A ) † = ( A † ) n (A^{n})^{\dagger} = (AA \ldots A)^{\dagger} = (A^{\dagger})^{n} ( A n ) † = ( AA … A ) † = ( A † ) n Let
where
A = − i H ( t 1 − t 2 ) ℏ . A = \frac{-iH(t_1 - t_2)}{\hbar}. A = ℏ − i H ( t 1 − t 2 ) . Hence,
A † = [ − i H ( t 1 − t 2 ) ℏ ] † = i H † ( t 1 − t 2 ) ℏ = i H ( t 1 − t 2 ) 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*} A † = [ ℏ − i H ( t 1 − t 2 ) ] † = ℏ i H † ( t 1 − t 2 ) = ℏ i H ( t 1 − t 2 ) = − A , where the fact that H = H † H=H^{\dagger} H = H †
U † = [ e A ] † = e A † = e − A . \begin{align*}
U^{\dagger} &= \big[ e^{A} \big]^{\dagger} \\
&= e^{A^{\dagger}} \\
&= e^{-A}.
\end{align*} U † = [ e A ] † = e A † = e − A . Hence,
U U † = e A e − A , = e A − A = e 0 , \begin{align*}
UU^{\dagger} &= e^{A}e^{-A}, \\
&= e^{A-A} \\
&= e^{0},
\end{align*} U U † = e A e − A , = e A − A = e 0 , where 0 is the zero operator. The operator expansion of the exponential of the zero operator then gives the identity,
e 0 = I . e^{0} = \mathbb{I}. e 0 = I . Repeating for U † U U^{\dagger}U U † U
One can therefore succinctly model the evolution of closed systems in quantum theory through unitary operators,
∣ ψ ′ ⟩ = U ∣ ψ ⟩ , U U † = U † U = I , \ket{\psi'} = U \ket{\psi}, ~ ~ UU^{\dagger} = U^{\dagger}U = \mathbb{I}, ∣ ψ ′ ⟩ = U ∣ ψ ⟩ , U U † = U † U = I , without concern for what the specific Hamiltonian is.
Properties of Unitary Operators ¶ Unitary operators have the following properties:
They act linearly on superpositionsLet ∣ + ⟩ = 1 2 ( ∣ 0 ⟩ + ∣ 1 ⟩ ) ∈ H \ket{+} = \frac{1}{\sqrt{2}} \big( \ket{0} + \ket{1}) \in \mathcal{H} ∣ + ⟩ = 2 1 ( ∣ 0 ⟩ + ∣ 1 ⟩ ) ∈ H U ∣ + ⟩ = 1 2 ( U ∣ 0 ⟩ + U ∣ 1 ⟩ ) U \ket{+} = \frac{1}{\sqrt{2}} \big( U\ket{0} + U\ket{1}) U ∣ + ⟩ = 2 1 ( U ∣ 0 ⟩ + U ∣ 1 ⟩ ) They take normalised states to normalised statesLet ∣ ψ ⟩ ∈ H \ket{\psi} \in \mathcal{H} ∣ ψ ⟩ ∈ H ⟨ ψ ∣ ψ ⟩ = 1 \braket{\psi|\psi} = 1 ⟨ ψ ∣ ψ ⟩ = 1 ∣ ψ ′ ⟩ = U ∣ ψ ⟩ \ket{\psi'} = U \ket{\psi} ∣ ψ ′ ⟩ = U ∣ ψ ⟩ ⟨ ψ ′ ∣ ψ ′ ⟩ = ⟨ ψ ∣ U † U ∣ ψ ⟩ = ⟨ ψ ∣ ψ ⟩ = 1 , a s U † U = U U † = I . \braket{\psi'|\psi'} = \bra{\psi}U^{\dagger}U\ket{\psi} = \braket{\psi|\psi}=1,~~{\rm as}~ ~ U^\dagger U=UU^\dagger =\mathbb{I}. ⟨ ψ ′ ∣ ψ ′ ⟩ = ⟨ ψ ∣ U † U ∣ ψ ⟩ = ⟨ ψ ∣ ψ ⟩ = 1 , as U † U = U U † = I . They represent reversible dynamics.If U U U V = U † V=U^\dagger V = U † U U U ∣ ψ ⟩ → ∣ ψ ′ ⟩ \ket{\psi} \rightarrow \ket{\psi'} ∣ ψ ⟩ → ∣ ψ ′ ⟩ V V V ∣ ψ ′ ⟩ → ∣ ψ ⟩ \ket{\psi'} \rightarrow \ket{\psi} ∣ ψ ′ ⟩ → ∣ ψ ⟩ V V V U U U