# 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.

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### Can the hamiltonian be derived from phase space evolution?

Given the phase space evolution of a system, $x(t)$ and $p(t)$, is there any way of getting the hamiltonian to make a later study of the system under the hamiltonian formalism? My first thought was ...
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We need to find the $g$ transformation's generator under a rotation on the phase space, where the Hamiltonian is equal to: $H = \frac{p^{2} + x^{2}}{2}$. At first i made the expression of an ...
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### The connection between classical phase space and quantum multiplicity

I am aware of the relationship $N = V/h^n$ where $N$ is the quantum multiplicity, $n$ is the number of position (or momenta) degrees of freedom, $V$ is the volume of classical phase space and $h$ is ...
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### Is the Wigner function a signed measure?

I have read in Wikipedia that quasiprobability distributions in phase space quantum mechanics may fail to be $\sigma$-additive, but I don't know in which sense this is true. If I have a Wigner ...
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### How do i read a pendulum phase diagram?

I'm trying to understand how the intermediate axis theorem works. And in one of the works that I found, they used a pendulum phase diagram, but idk how to read it. Can anybody help please? The work ...
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### What is the difference between a macrostate and multiplicity?

The entropy of a system of an ideal gas depends on the external parameters $U, V, N$. I always thought entropy is defined by a certain macrostate, which is a set of given external conditions like ...
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### Why does the number of accessible microstates always increase?

If we consider two systems with number of accessible microstates A and B and internal energies of E(A) and E(B) that are exchanging an small amount of heat Q from A to B we get the following: A total ...
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### What is the analogue for symplectic structure in case of spin variables?

According to some (e.g. Haroche and Raimond in Exploring the quantum: atoms, cavities and photons), the quantum world consists (mainly) of spins and harmonic oscillators. For harmonic oscillators (i.e....
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### Hamilton-Jacobi Equation & Canonical Transformation

I am attempting to solve the Hamilton-Jacobi Equation in the case of a simple harmonic oscillator, to recover the associated generating function and the generated canonical transformation. Consider ...
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### What are the Fock-state probabilities of general Gaussian states?

A general (pure) Gaussian state has the form $\newcommand{\on}{\operatorname{#1}}\newcommand{\ket}{\lvert #1\rangle}\ket{\alpha,\xi}\equiv D(\alpha)S(\xi)\ket{\on{vac}}$, with $\ket{\on{vac}}$ ...
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### The “real butterfly effect”

This question stems from the confusion that I feel after reading this popular blog post by Sabine Hossenfelder. It is based on this paper which is paywalled, unfortunately. The claim is the following: ...
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### Why does the separatrix in phase portraits have infinite period and pass through at least one unstable equillibrium point?

In the case of 1D Hamiltonians not explicitely dependent on time, our professor claims that the "period" of the separatrix is necessarily infinite and must pass through an unsable ...
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### Composite systems [closed]

Studying QM attracted my attention to the concept of a composite system. I erroneously thought, following classical reasoning, that a composite system can be i.e. a system formed by two qubits and ...
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### How is Hamilton's first equation useful in solving mechanics problems? [duplicate]

Here is the first Hamilton equation: $\frac{\partial H}{\partial {p}_q} = \dot{q}$ Let's use it. Imagine a ball rolling down a frictionless hill (ignore the friction vector in the image). As time goes ...
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### Addition of a constant to the operator due to quantization

Groenewold in his book On the Principles of Elementary Quantum Mechanics (1946, Springer Netherlands) page 45, maps the canonical momentum $p^2$ in the classical phase space to a general canonical ...
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### Name for region in phase space with no outgoing or incoming flows?

I've been looking for a term online but couldnt find it: suppose we have a subset $X$ in phase space, such that for all $q\in X$, the path starting at $q$ never exits $X$ either forward or backward in ...
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### Liouville's theorem for the submanifold of given conserved quantities?

Liouville's theorem states that phase space volume is conserved over time with respect to the dynamical system generated by the Hamiltonian and Hamilton's equations. However, any given point in phase ...
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### Rovelli's relativistic phase space

I'm asking for help to understand the definition of relativistic phase space given by Rovelli in his book Quantum Gravity. At chapter 3, he states those following definitions The relativistic phase ...
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### What is wrong with Weyl-Wigner representation?

The Weyl-Wigner representation is a useful tool to study QM from a semiclassical, phase-space point of view. My question is simple: if this method is so close to classical mechanics, why don't we use ...
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### Liouville's Theorem derivation

As an example of the book "Introduction to Quantum Mechanics Schrodinger Equation and Path integral" by Harald J. W. Muller. We have to prove Liouville's theorem. Here I show the proof as ...