Dynamics

Describing the system at a given time is not enough, of course: our theory must also describe how the state of the system evolves with time. We would expect at least, for each time t, a map $\alpha_t$ on the set of all states, which takes a state at time s into the state at time s+t. These maps should satisfy the relations  

$$\alpha_0 = I \qquad \mbox{and} \qquad \alpha_t \alpha_s = \alpha_{s+t} .$$ (5)

(Note that these equations imply that $\alpha_t \alpha_{-t} = \alpha_0 = I$, so that each $\alpha_t$ is a bijection.) There is one other natural requirement: since each $\alpha_t$ is a bijection, the system will be in state $\alpha_t(\tilde{\psi})$ at time s+t if and only if it is in state $\tilde{\psi}$ at time s. It follows that the probability the probability of the system being in state $\alpha_t(\tilde{\phi})$ given that it is in state $\alpha_t(\tilde{\psi})$ should be independant of t, so we would expect our evolution operators to preserve the ray product (4):  

$$\left[ \alpha_t (\tilde{\phi}), \alpha_t (\tilde{\psi}) \right] = \left[ \tilde{\phi}, \tilde{\psi} \right] .$$ (6)

These assumptions are all we need for the first form of our dynamical postulate:

(P3) (First version) For each time t, there is a map $\alpha_t$, defined on the set of all states, such that if the system is in state $\tilde{\psi}$ at time s, it is in state $\alpha_t(\tilde{\psi})$ at time s+t. The maps $\{ \alpha_t \}_{t \in {{\rm I}\kern-.18em{\rm R}}}$ satisfy (5) and (6).

This form of the dynamical postulate seems well motivated, but it is undeniably abstract: most of us are not used to thinking in terms of maps whose domains are sets of subspaces of a Hilbert space! Fortunately, this is not necessary: it is a consequence of a theorem of Wigner and Bargmann that the maps $\alpha_t$, which are defined on states, can be `lifted' to linear operators U(t) on ${\cal H}$ itself, in such a way that

$$\alpha_t (\tilde{\psi}) = \widetilde{U(t)\psi} .$$

It is even possible to require that the maps U(t) are unitary:  

$$\langle U(t)\psi ,U(t)\psi\rangle = \langle \psi ,\psi\rangle;$$ (7)

and that the collection $\{ U(t) \}_{t \in {{\rm I}\kern-.18em{\rm R}}}$ forms a group:  

$$U(0) = I \qquad \mbox{and} \qquad U(t) U(s) = U(s+t) .$$ (8)

Any collection $\{ U(t) \}_{t \in {{\rm I}\kern-.18em{\rm R}}}$ of operators satisfying (7) and (8) is called a 1-parameter unitary group. We can now state our second (equivalent) form of the dynamical postulate:

(P3) (Second version) The time evolution of the system is governed by a 1-parameter unitary group $\{ U(t) \}_{t \in {{\rm I}\kern-.18em{\rm R}}}$, in such a way that if the vector $\psi \in {\cal H}$ is a representative of the state of the system at time s, then the state at time t+s is represented by $U(t)\psi$.

At this point, a miracle occurs! It turns out that 1-parameter unitary groups are well understood. By Stone's Theorem, any such group has an infinitesimal generator A, a self-adjoint operator in ${\cal H}$ such that  

$$U(t) = e^{\imath tA} .$$ (9)

(If A is bounded--which it usually is not--the exponential function here can be defined by the usual power series; in the general case, the definition requires the Spectral Theorem again.) The infinitesimal generator of the time-translation group of a quantum system is called the Hamiltonian of the system, and is usually denoted by H. Since H is self-adjoint, it is a candidate for an observable, and it is usually interpreted as the total energy of the system.

Finally, we observe that applying (9) (with A replaced by H) to a vector $\psi$ and differentiating with respect to t gives  

$$\frac{d}{dt} \psi(t) = \imath H\psi(t)$$ (10)

(where we have written $\psi(t) := U(t)\psi$.) This is the Schrödinger equation, the starting point of `wave mechanics', one of the early forms of quantum theory. (The other was Heisenberg's `matrix mechanics'; they were shown to be equivalent by Dirac.)