Schrödinger Equation – Generality of the Schrödinger Equation in Quantum Mechanics

hilbert-spaceoperatorsquantum mechanicsschroedinger equationtime evolution

According to the Schrödinger equation
$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$
the transformation $U_t:\mathcal H\to \mathcal H: \Psi\mapsto \Psi(t)=e^{-itH}\Psi$ for every $t\in\mathbb R$ is a unitary transformation of the Hilbert space $\mathcal H$, and since from this equation,

$$U_tU_s=U_{t+s}\tag 2,$$
the mapping, $t\mapsto U_t$ it is a one-parameter subgroup of the unitary group.

Starting from the other direction, if we suppose that the evolution of the states is a one-parameter subgroup of the unitary group, Stone's theorem yields (1).

But, since quantum mechanical states are not elements of $\mathcal H$, but they are elements of the projective Hilbert space $\mathcal P(\mathcal H)$, time evolution isn't a one-parameter subgroup of the unitary group, but it is a one-parameter subgroup of the projective unitary group, that is, by Wigner's theorem, instead of (2), only
$$U_tU_s=\omega(t,s)U_{t+s}\tag 3$$
must hold, where $\omega(t,s)$ is a complex number of modulus $1$ depending on $t$ and $s$.

From (2), Stone's theorem yields (1), but from (3), what theorem yields what?

Best Answer

There are two theorems relevant here. The former proves that actually, under natural hypotheses, the multipliers $\omega$ can be removed and one can safely apply the Stone theorem. This theorem is an immediate corollary of the Bargmann theorem as the Lie group $\mathbb{R}$ is (simply connected and) Abelian and its Lie algebra cohomology is trivial, or also can be directly proved (it was established by Wigner independently).

  1. (Wigner-Bargmann) If $\mathbb{R}\ni t \mapsto U_t$ is a projective-unitary (so there are your $\omega$) representation such that $|\langle \psi|U_t\phi\rangle|$ is a continuous function of $t$ for every choice of $\psi,\phi$ in the Hilbert space, then it is possible to re-arrange the multipliers $\omega$ (by multiplying the $U_t$ with suitable phases $U’_t:= e^{if(t)}U_t$) in order to obtain a strongly-continuos properly unitary representation.

The latter shows that, if the representation is unitary, continuity is almost automatic, since non Borel-measurable functions are really difficult to find.

  1. (von Neumann) if $\mathbb{R} \ni t \mapsto \langle \psi| U_t\phi\rangle$ is Borel measurable for every choice of $\psi,\phi$ of a separable Hilbert space where each $U_t$ acts as a unitary operator and the $U_t$ form a group representation of $\mathbb{R}$, then $t\to U_t$ is strongly continuous. So we can apply the Stone theorem.
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