2.1 Basics about quantum mechanics

Let us recall the basics. Given a quantum system, we have a Hilbert space \(\mathcal{H}\), in which a state lives. A unit state \(\ket{\psi}\) and another \(\ket{\psi'}\) describes the same state, which we write as \(\ket{\psi}\sim\ket{\psi'}\), if (and only if) \[\begin{equation} \tag{2.1} \ket{\psi} = e^{\mathrm{i}\alpha}\ket{\psi'} \end{equation}\] for some phase \(\alpha\). Then, we define the ray space \(\mathbb{P}\mathcal{H}\) as the projective space of \(\mathcal{H}\): \[\begin{equation} \tag{2.2} \mathbb{P}\mathcal{H} := S\mathcal{H}/\sim, \end{equation}\] where \(S\mathcal{H}\) denotes the space of unit states. By definition, elements in the ray space are one-to-one corresponded to physical states of the system. Given two states \([\ket{\psi}],[\ket{\phi}] \in \mathbb{P}\mathcal{H}\), where \([,]\) denotes the equivalence class in the definition (2.2), the transition probability of the two states are \[\begin{equation} \tag{2.3} \lvert \braket{\psi|\phi}\rvert^2. \end{equation}\] Note that this definition does not depends on the choice of the representatives \(\psi,\phi\) in each class. We put the structure of abelian group to \(C^2(G,M)\) induced by the abelian group structure of \(M\).

2.1.1 operator algebra*

Sometimes it is convenient to focus on operators acting on the states, rather than the states itself. We let the algebra of the (bounded) operators be denoted by \(\mathcal{A}\). If the Hilbert space is finite dimensional, i.e. \(\mathcal{H} = \mathbb{C}^n\) for some integer \(n\), the algebra is simply the matrix algebra: \[\begin{equation} \tag{2.4} \mathcal{A} \cong \mathrm{Mat}(\mathbb{C}^n). \end{equation}\]

In this lecture we consider the dynamics by a constant Hamiltonian3. We use the Heisenberg picture, so the state \(\ket{\psi}\) does not develop but an operator \(\mathcal{O}\) develops in time as \[\begin{equation} \tag{2.5} \mathcal{O}(t) = e^{\mathrm{i}H t} \mathcal{O}(0) e^{-\mathrm{i}H t}. \end{equation}\] Here we have set the plank constant \(\hbar =1\).

  1. Instead, one might consider a discrete (finite time) evolution by a unitary operator. Such a system is often called Floquet system. The term originally means that you have an external field periodic in the time. The discrepancy between the continuous and discrete evolutions is an interesting ongoing research topic.↩︎