nLab
quantum mechanics

under construction

Quantum mechanics of point particles

Quantum mechanics is usually understood to be the part of quantum field theory that studies the quantum analog of the classical mechanics of point particles.

Zoran: While this has some merit for this particular direction/subfield, it is other way around: quantum field theory is a special kind of a quantum mechanical system (with infinitely many degrees of freedom). You are here taking very special point of view and identifying quantum mechanics with quantum mechanics of a system with finitely many degrees of freedom. One teaches thus other way around: there are quantum mechanical systems, then there are some special kinds including QFTs. Quantum mechanics of the point particles on the other hand is not subsumed by the formalism below either. For example, the time-dependent Hamiltonians and quantum mechanics for finite state systems do not seem to be included. I personally do not understand how do you put a concrete potential in the game below either but I suppose it is a way. I mean which space-dependent and other parameters are allowed for $H$ and how it works (note that while the usual forces are related to the curvature, I do not see there is a mechanism for all kind of potentials and more general hamiltonians to be just derived from general rule and be invariant under isomorphism in Riemannian category) ?

Toby: It looks like Urs understands ‘mechanics’ in much the same way that I did at classical mechanics. I do agree that that the term includes time-dependent Hamiltonians and similar generalisiations, however.

One may usefully think of the quantum mechanics of a point particle propagating on a manifold $X$ as being $(0+1)$-dimensional quantum field theory:

the fields of this system are maps $\Sigma \to X$ where $\Sigma \in \mathrm{Riem}{\mathrm{Bord}}_1$ are 1-dimensional Riemannian manifold cobordisms. These are the trajectories of the particle.

After quantization this yields a 1-dimensional FQFT given by a functor

$$U(-):\mathrm{Riem}{\mathrm{Bord}}_1\to \mathrm{Hilb}$$U(-) : Riem Bord_1 \to Hilb

from cobordisms to Hilbert spaces (or some other flavor of vector spaces) that assigns

• to the point the space of states $\mathscr{H}$, typically the space of $L_2$-sections (with respect to a Riemannian metric on $X$) of the background gauge field on $X$ under which the particle in question is charged

• to the cobordism of Riemannian length $t$ the operator

  $$U(t):=\exp \left(\frac{t}{i\hslash }H\right):\mathscr{H}\to \mathscr{H},$$U(t) := \exp\left(\frac{t}{i \hbar } H \right) : \mathcal{H} \to \mathcal{H} \,,

  where $H$ is the Hamiltonian operator, typically of the form $H={\nabla }^{†}\circ \nabla$ for $\nabla$ the covariant derivative of the given background gauge field.

Such a setup describes the quantum mechanics of a particle that feels forces of backgound gravity encoded in the Riemannian metric on $X$ and forces of background gauge fields (such as the electromagnetic field) encoded in the covariant derivative $\nabla$.

Quantum physics in general

More generally, quantum mechanics or quantum physics may be taken to subsume quantum field theory and all of physics that is not classical mechanics. Another way to look at quantum processes is via quantum channels which are completely positive trace-preserving maps.

Zoran: I slightly disagree with that opposite extremity either. The sentence is better suited to define quantum physics (though complement to classical physics includes relativity, while opposite to classical mechanics includes thermodynamics). Quantum physics is not the same as quantum mechanics. There are quantum phenomena which are not treated by quantum mechanics only. I mean it would be very unusual calling quantum statistical physics, part of quantum mechanics; it requires special limiting assumptions (thermodynamic limit, quantum ergodicity and so on lie outside) outside of scope of the things derivable from quantum mechanics.

Ian Durham: J.J. Sakurai’s book Modern Quantum Mechanics (not to be confused with his Advanced Quantum Mechanics which is a field theory book) discusses quantum statistical mechanics under the guise of quantum mechanics as does the forthcoming book Q-PSI: Quantum processes, systems, and information by Schumacher and Westmoreland. While it may not be entirely accurate, “quantum physics” and “quantum mechanics” are usually treated as being synonymous.

Quantum mechanics in terms of $†$-compact categories

Many aspects of quantum mechanics and quantum computation depend only on the abstract properties of Hilb characterized by the fact that it is a †-compact category.

For more on this see

• quantum mechanics in terms of †-compact categories.

Related entries

• Hamilton operator, semiclassical approximation