3.6 Some quantum dynamics

The time evolution of a quantum state is a unitary process which is generated by a Hermitian operator called the Hamiltonian, which we also59 denote by H. The Hamiltonian contains a complete specification of all interactions within the system under consideration. In an isolated system, the state vector |\psi(t)\rangle changes smoothly in time according to the Schrödinger equation: \frac{\mathrm{d}}{\mathrm{d}t} |\psi(t)\rangle =-\frac{i}{\hbar} H |\psi(t)\rangle.

For time-independent Hamiltonians (i.e. where |\psi(t)\rangle=|\psi\rangle has no t-dependence), the formal solution of this reads60 \begin{gathered} |\psi(t)\rangle = U(t)|\psi(0)\rangle \\\quad\text{where}\quad U(t) = e^{-\frac{i}{\hbar}Ht}. \end{gathered}

Now, to relate this to the earlier parts of this chapter, we note that the Hamiltonian of a qubit can always be written in the form H = E_0\mathbf{1}+\omega(\vec{n}\cdot\vec{\sigma}), hence \begin{aligned} U(t) &= e^{-i\omega t \vec n\cdot\vec\sigma} \\&= (\cos\omega t)\mathbf{1}- (i\sin\omega t)\vec{n}\cdot\vec{\sigma} \end{aligned} which is a rotation with angular frequency \omega about the axis defined by the unit vector \vec n.

!!!TO-DO!!! the 4\pi world of qubits

  1. Hopefully it will always be clear from the context which H refers to Hamiltonian and which H to Hadamard. Don’t confuse the two!↩︎

  2. Here \hbar denotes Planck’s constant, which has the value \hbar = 1.05\times 10^{-34}\,\text{ Js}. However, theorists always choose to work with a system of units where \hbar = 1.↩︎