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Questions tagged [phase-space]

A notional even-dimensional space representing all relevant states of a dynamical system; it normally consists of all components of position and momentum/velocity involved in that unique specification. Use for both classical and quantum physics.

A notional even-dimensional space representing all relevant states of a dynamical system; it normally consists of all components of position and momentum/velocity involved in that unique specification. Use for both classical and quantum physics.

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What is the "secret " behind canonical quantization?

The way I (and perhaps most students around the world) was taught QM is very weird. There is no intuitive explanations or understanding. Instead we were given a recipe on how to quantize a classical theory, which is based on the rule of transforming…
Jacob
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How is Liouville's theorem compatible with the Second Law of Thermodynamics?

The second law says that entropy can only increase, and entropy is proportional to phase space volume. But Liouville's theorem says that phase space volume is constant. Taken naively, this seems to imply that the entropy can never change. What's…
knzhou
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Is there a physical system whose phase space is the torus?

NOTE. This is not a question about mathematics and in particular it's not a question about whether one can endow the torus with a symplectic structure. In an answer to the question What kind of manifold can be the phase space of a Hamiltonian…
joshphysics
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Why does non-commutativity in quantum mechanics require us to use Hilbert spaces?

I am reading Why we do quantum mechanics on Hilbert spaces by Armin Scrinzi. He says on page 13: What is new in quantum mechanics is non-commutativity. For handling this, the Hilbert space representation turned out to be a convenient — by many…
Stan Shunpike
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Sympletic structure of General Relativity

Inspired by physics.SE: Does the dimensionality of phase space go up as the universe expands? It made me wonder about symplectic structures in GR, specifically, is there something like a Louiville form? In my dilettante understanding, the existence…
genneth
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What is the difference between configuration space and phase space?

What is the difference between configuration space and phase space? In particular, I notices that Lagrangians are defined over configuration space and Hamiltonians over phase space. Liouville's theorem is defined for phase spaces, so is there an…
Alexander
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The geometrical interpretation of the Poisson bracket

"Hamiltonian mechanics is geometry in phase spase." The Poisson bracket arises naturally in Hamiltonian mechanics, and since this theory has an elegant geometric interpretation, I'm interested in knowing the geometrical interpretation of the…
Samà
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Why isn't momentum a function of position in quantum mechanics?

In quantum mechanics, the unitary time translation operator $\hat{U}(t_1,t_2)$ is defined by $\hat{U}(t_1,t_2)|ψ(t_1)\rangle = |ψ(t_2)\rangle$, and the Hamiltonian operator $\hat{H}(t)$ is defined as the limit of $i\hbar\frac{\hat{U}(t,t+\Delta…
Keshav Srinivasan
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The quantum analogue of Liouville's theorem

In classical mechanics, we have the Liouville theorem stating that the Hamiltonian dynamics is volume-preserving. What is the quantum analogue of this theorem?
poisson
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What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics?

One of the most important results of Classical Mechanics is Liouville's theorem, which tells us that the flow in phase space is like an incompressible fluid. However, in the phase space formulation of Quantum Mechanics, one of the main results due…
jak
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Which transformations are canonical?

Which transformations are canonical?
Kishor Bharti
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Topology of phase space

Context: From Liouville's integrability theorem we know that: If a system with $n$ degrees of freedom exhibits at least $n$ globally defined integrals of motion (i.e. first integrals), where all such conserved variables are in Poisson involution…
user929304
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Question about canonical transformation

I was going through my professor's notes about Canonical transformations. He states that a canonical transformation from $(q, p)$ to $(Q, P)$ is one that if which the original coordinates obey Hamilton's canonical equations then so do the…
Dargscisyhp
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Integrable vs. Non-Integrable systems

Integrable systems are systems which have $2n-1$ time-independent, functionally independent conserved quantities ($n$ being the number of degrees of freedom), or $n$ whose Poisson brackets with each other are zero. The way I understand it, these…
Ronak M Soni
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Understanding "natural variables" of the thermodynamic potentials using the example of the ideal gas

I'm struggling with the concept of "natural variables" in thermodynamics. Textbooks say that the internal energy is "naturally" expressed as $$ U = U(S,V,N)$$ For an ideal gas, I could take the Sackur–Tetrode equation - which gives me $S(U,V,N)$ -…
Marc
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