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On page 46 of Sarkurai's Modern QM, he defined momentum as the generator of infinitesimal translation of a QM system. Later with similar methods, he defined the generator of time evolution of QM system is $H$ (not to be mixed up with the classical definitions of H, which is the time invariance of $L$agriange).

However, at page 85,86, he suddenly write $H=\frac{p^2}{2m}+V$ without any proofs.

My question: Are there any ways to prove that $\frac{p^2}{2m}+V$ is the time invariance in (a type of) QM system?


I am still trying to understand the first 100 pages of Sarkurai's QM. I could solve most of the problems on intro functional analysis textbook but why QM seems so hard?

Please correct any false understandings of mine.

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  • $\begingroup$ I haven't looked through the book, but before he proved it was the time evolution operator did he not prove that H was the energy operator? Or defined it as such, or derived that it is such from its definition though the Lagrangian? If he did then it's kinetic plus potential energy and you're set. He needed to have defined H at some point. $\endgroup$ – Bob Bee Jun 19 '17 at 3:50
  • $\begingroup$ Comment to the post (v1): Noether's theorem deals with classical (as opposed to quantum) systems with an action formulation. For energy conservation of a classical Lagrangian system, see physics.stackexchange.com/q/94381/2451 and links therein. $\endgroup$ – Qmechanic Jun 19 '17 at 4:39
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In nonrelativistic quantum mechanics, you can't really derive the specific form of the Hamiltonian from first principles - it's usually just taken to be part of the definition of the system, just as it is for classical mechanics. While logical causation is always a bit of a philosophical issue in physics, we usually think of the Hamiltonian as determining the system's time evolution, not the other way around.

If you know how the system behaves in the classical limit, then you can take the classical Hamiltonian and quantize it by promoting the classical phase-space variables to noncommuting quantum operators (although this procedure can be ambiguous for complicated Hamiltonians). Empirically, we know that many nonrelativistic single-particle classical system are well described by Hamiltonians of the form $H = p^2/(2m) + V(x)$ for some potential energy function $V$, and this empirical pattern is what guides us to consider quantum Hamiltonians of the same form. But this certainly isn't the only possible quantum Hamiltonian that comes up in the real world - for example, the kinetic energy term gets modified in the presence of a magnetic field.

In relativistic QM, theories are usually defined by specifying a Lorentz-invariant Hamiltonian (or Lagrangian) density, which in turn spatially integrates to the full Hamiltonian - a very different formalism from the nonrelativistic case. In this regime, we do have a "master Hamiltonian" - that of the Standard Model - which we believe accurately describes virtually all possible systems in which the effects of gravity aren't important. Unfortunately, the Standard Model Hamiltonian is far too complicated to directly describe any processes more complicated than few-particle scattering.

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There are ways to study the relation between self-adjoint Hamiltonian operators and unitary evolutions. Mathematically, there is a one-to-one correspondence between strongly continuous unitary representations of the reals (seen as an abelian group) $(U(t))_{t\in\mathbb{R}}$ and self-adjoint operators on the Hilbert space $\mathscr{H}$ of the representation.

So given any self-adjoint operator of the form $H=-\Delta +V(x)$ with domain $D(H)$ on the Hilbert space $L^2(\Omega)$, $\Omega\subseteq \mathbb{R}^d$, there is one strongly continuous representation of the reals generated by it, and often denoted by $(e^{-i\frac{t}{\hslash}H})_{t\in\mathbb{R}}$. So considering the dynamical quantum system $\Bigl[\mathscr{B}\bigl(L^2(\Omega)\bigr),(e^{-i\frac{t}{\hslash}H}\,\cdot\, e^{i\frac{t}{\hslash}H})_{t\in\mathbb{R}}\Bigr]$, composed of an algebra of observables (the continuous operators on the Hilbert space) and an automorphism of time evolution, the observable $H$ generates the dynamics and it is left invariant by it.

Conversely, given a dynamical system $[\mathcal{W}, (U(t)\,\cdot\,U^*(t))_{t\in\mathbb{R}}]$ where $\mathcal{W}\subseteq \mathscr{B}(\mathscr{H})$ is a W*-algebra and $(U(t))_{t\in\mathbb{R}}$ is a strongly continuous group of unitary operators on $\mathscr{H}$, then the generator of time evolution, left invariant by it, is given by $H=\frac{d}{dt} U(t)\Bigr\rvert_{t=0}$ (where the derivative is taken in the strong operator sense in a suitable dense domain). $H$ is a self-adjoint operator on $\mathscr{H}$.

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