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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Translating Hamilton's equations in Phase Space trajectories

I'm trying to understand how I can visualize the phase space of a Hamiltonian based on Hamilton's equations. My Hamiltonian is: $H=\lambda xp$. $$\dot{x} = \lambda x$$ $$\dot{p} = -\lambda p$$ $$x= ...
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Need help finding Hamiltonian for equations of motion

I have the following equation of motion: $$\ddot \theta+\dot\theta^2\theta+k^2\theta=0.\tag 1$$ This equation is from this question. I wanted to see if I could find a Hamiltonian for this equation but ...
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Is a damped, driven simple harmonic oscillator a limit cycle?

I've been reading about limit cycles and synchronization from Pikovsky's Synchronization in order to build a background for non-linear oscillators. What I know about the limit cycles is that they're ...
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Canonical variables

In this paper Quantum geometric tensor and quantum phase transitions in the Lipkin-Meshkov-Glick model in Eq.(9), the author express the Hamiltonian as $$H = -\Omega \cos\theta+\Omega_x\sin\theta\cos\...
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Why do $\{Q,P\}$ form an irreducible set for a particle with no internal degrees of freedom?

In deriving expressions for the generators of Galilean symmetries $\mathbf{J}, \mathbf{P}, \mathbf{G}, H$, Ballentine (Quantum Mechanics: A Modern Development) uses that $\{\mathbf{Q},\mathbf{P}\}$ ...
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Intuitively, why does Quantum Mechanics involve a sum over all possibilities?

I understand that one can just mathematically derive the path integral from the Schrodinger equation. I'm looking for an intuitive explanation in contrast with classical mechanics. Consider a ...
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Constraints on Phase Space

This question here motivated me to record to the following fact: Consider a $2n$ dimensional phase space with coordinates $q_1,...,q_n,p_1,...,p_n$. Consider the constraint $C(\vec q)=0$. What is the ...
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Proof of the valence $\lambda$ of a canonical transformation equaling its Jacobian determinant [closed]

Let $Q(q,p),P(q,p)$ be a canonical transformation with valence $\lambda$. The following is intended to be a proof of the following relation: $$\lambda = \frac{\partial(Q,P)}{\partial(q,p)}.$$ Let $F(q,...
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"Constrain then quantise" vs. "quantise then constrain"

Consider a classical system whose configuration space is a manifold $M$, and which is subject to some constraint $\mathcal{C}=0$. [E.g. the system could be a particle moving in $M=\mathbb{R}^n$, with $...
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Derivation of Hamilton-Jacobi (HJ) Equation

In the Derivation of Hamilton Jacobi Equation, I didn't understand the bold parts: we can write (1) formally as, $$ \frac{\partial F\left(q_i, Q_i, t\right)}{\partial t}=-H\left(p_i, q_i, t\right)=-H\...
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Is there some notion of classical uncertainty which quantizes to quantum uncertainty?

I would like to know if there is some notion of classical uncertainty which quantizes to give quantum uncertainty? For instance, suppose we have a classical system whose phase space is given by a ...
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Question about deriving the amount of pressure due to an element of momentum space in cosmology (Baumann Cosmology Book Eq. 3.10)

In Daniel Baumann's cosmology book Eq. (3.10) and cosmology lecture notes Eq. (3.2.18) he states that the pressure in the early universe can be defined as $$P=\frac g{(2\pi)^3}\int d^3 p\ f(p)\times\...
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What are good books/chapters of books or articles to study canonical transformations in quantum mechanics at a graduate level?

I am looking for any kind of sources about canonical transformations in quantum mechanics in the operatorial formulation of the theory and its connection with the classical canonical transformation ...
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What should be the definition of a comoving frame in phase space?

In short, I think there are two types of comoving frame when talking about distribution function, since it is defined in phase space. Which one should be the real one? I suppose this question is ...
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Is there a canonical Taylor expansion for operators in terms of $X$ and $P$?

Consider the algebra of operators acting on wavefunctions ($L^2(\mathbb{R})$) generated by $X$ and $P = -i\hbar (\partial/\partial x)$. For some operator $A$ in this algebra, or possibly in a ...
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Existence Of Phase Flow Provided Potential Energy is Positive

I am reading through Arnold's "Mathematical Methods Of Classical Mechanics". In the section 4D on p. 21 concerning Phase Flow there is a question that reads as follows: Show that if ...
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How to interpret Liouville's equation?

In various texts, I see Liouville's theorem stated both verbally and as an equation. But it seems to me that these two formulations don't agree. For example, in Wikipedia: The distribution function ...
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Poisson brackets for a field theory

I'm performing a calculation involving Dirac constraints theory, and I need to calculate the Poisson brackets between constraints and the total Hamiltonian. The starting theory is described by a ...
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Is there a relationship between the phase space path integral and phase space quantum mechanics?

I understand that they're, in the end, related because they're the same theory. But is there a closer relationship because both are theories of probability distributions on phase space? I also ...
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Why do we care about the canonical commutation relations?

Suppose $\hat{x}$ and $\hat{p}$ are the position and momentum operators, it can be shown that $$[\hat{x}, \hat{p}] = i\hbar\mathbb{I}.$$ The Stone-von Neumann theorem tells us that that the above is ...
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Quantum phase space volume conversion

Suppose we have some ensemble specified by the energy $E,$ particle number $N$, and volume $V.$ A classical phase space surface area can be calculated from this: $$\mathcal{S} = \int^{'} d \mathcal{V}=...
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Measure-preserving vs. non-dissipative vs. Hamiltonian systems

Are measure-preserving systems always non-dissipative? Phase space volume is preserved in both, according to Liouville's theorem. But are there any differences? Are measure-preserving and non-...
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Canonical Transformation of Poisson Bracket [closed]

In Goldstein section 9.4(pg 381) it tells us that for a Hamiltonian that is not explicitly time dependent, transformations of $Q = Q(q,p), P = P(q,p)$ are canonical if $$\frac{\partial Q}{\partial q} =...
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Can Quantum observables be modeled as real functions on the phase space?

In the context of justifying the failure of modeling quantum observables in the 'more natural' way as real functions on the phase space (i.e. similar to the mathematical image modeling classical ...
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How can you read from a phase space diagram how quickly a path gets traversed?

I'm learning about Hamiltonian mechanics and it is quite interesting. However I'm trying to understand how to see how quickly a path in phase space gets traversed. How do you read from a phase space ...
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What is the mathematical structure behind the statistical physics? [closed]

It is well known (relatively) that the classical mechanics (especially the analytical mechanics) can be formulated as vector fields on the tangent bundle (Lagrangian formulation) or cotangent bundle ...
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Volume in a scattering process equal to 1?

I'm studying Fermi's golden rule, and in section 8.3.1 of "Braibant, Giacomelli, Spurio - Particles and Fundamental Interactions" there is an application to the decay of the neutron. While ...
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Proof of Liouville's Theorem

The Wikipedia article on Canonical Transformations has a section on Liouville's Theorem. It makes the following argument: $$J=\frac{\partial(\mathbf{Q},\mathbf{P})}{\partial(\mathbf{q},\mathbf{p})}$$ ...
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Canonical Transformation "Indirect Approach"

The Wikipedia article on Canonical Transformations, in the section "Indirect Approach," makes the following argument: $$ \dot{Q}_{m} = \frac{\partial Q_{m}}{\partial \mathbf{q}} \cdot \dot{\...
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Connection between parameter space and configuration space

I am just wondering what is the connection between parameter space and configuration space (or phase space)? I know the connection between configuration space and phase space but it seems like any ...
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Equivalence of symplectic condition and canonical transformation

In Goldstein's "Classical Mechanics", at page 384 it is claimed that given a point-transformation of phase space $$\underline{\zeta} = \underline{\zeta}(\underline{\eta}, t),\tag{9.59}$$ ...
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Can the Einstein tensor be written as an integral over real spacetime?

Background We know the Stress-energy tensor can be written as: $$ T^{\mu \nu} = \int \mathcal{N}(x,p,t) p^\mu \otimes p^\nu \frac{d V_p}{E}$$ where $\mathcal{N}(x,p,t)$ is the distribution function, $...
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Phasespace density of $N$ harmonic oscilators

For one classical harmonic oscillator with Hamiltonian $$H = \frac{p^2}{2m}+\frac{m\omega^2}{2}x^2$$ the density of states can be calculated as by calculating the number of states with Energy smaller ...
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Wigner transform, convolution, and poles

Let \begin{equation} \int\mathrm{d}z~ A(x,z) B(z,y) = \delta(x - y). \end{equation} Taking Wigner transform of both sides we readily obtain \begin{equation} A^W(X,p) \star B^W(X,p) = 1, \end{equation} ...
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Given a system, how to decide whether a closed orbit is homoclinic, not periodic, solely based on its phase portrait?

Background and definitions: A system is conservative if it has at least one conserved quantity. In a phase portrait of a nonlinear conservative system, trajectories that start and end at the same ...
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Shape Preservation Under Canonical Transformations

It is well known that under a canonical transformation, the differential volume is conserved (The Integral Invariants of Poincare). The density of states in an ensemble of systems obeying the same ...
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Difference between Symmetric and Phase difference?

For a simple coupled oscillator system such as the one here, with equal spring constants and equal masses (with a displacement from equilibrium of $x_1$ and $x_2$), it follows that: $(\ddot{x}_1+\ddot{...
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How to integrate a system with a hamiltonian with non-diagonal kinetic energy

I have the following classical Hamiltonian for two coupled oscillators in the same molecule: $$H=T+V =\left(\frac{p_1^2}{2\mu}+\frac{p_2^2}{2\mu}+k_pp_1p_2\right)+ \left(\frac{1}{2}\mu\omega_1^{2}x_1^...
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What is the most general wave function of a minimum uncertainty (Gaussian) state in quantum mechanics?

For some state $|\psi\rangle$ it is possible to recover the uncertainty principle using the fact that $$\left|(\hat{\sigma_{Q}}-i\lambda\hat{\sigma_{P}})|\psi\rangle\right|^{2}\geq0,$$where$$\hat{\...
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Why does physics need high-dimensional spheres? [closed]

Under what physical scenarios do the properties of high-dimensional spheres come in handy? If physics is 3 or 4-dimensional, when does one need to interest oneself in spheres of dimensionality higher ...
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Non-uniqueness of Glauber-Sudarshan $P$-function

For a state $\rho$ acting on single bosonic mode with coherent states $|\alpha\rangle$, one can always define a $P$-function to furnish a diagonal representation of the state in the coherent-state ...
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Do we lose information about the state when we obtain the Wigner function by solving the eigenvalue equation?

It can be shown that $$H(q,p)\star W_{\psi}(q,p)=EW_{\psi}(q,p)$$ where $H(q,p)$ is the classicaly Hamiltonian function, $\star$ is the Moyal/Groenewold star product and $W_{\psi}(q,p)$ is the Wigner ...
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Coordinate transformation of an expression

I have an expression for $[F,G]_{\mathbf{p},\omega,\mathbf{R},T}$ as $$ [F,G]_{\mathbf{p},\omega,\mathbf{R},T} = \frac{\partial F}{\partial \omega} \frac{\partial G}{\partial T} -\frac{\partial F}{\...
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What if we skip polarization in geometric quantization?

In QM, we most frequently work with "position-space" representation of the CCR $$ \mathcal{H} = L_2(\mathbb{R}, dx), \quad X = x, \quad P = - i \hbar \frac{d}{dx}. $$ Sometimes it's useful ...
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What is a phase space?

What is a phase space? And can the phase space be specified with x and y instead of with theta and omega? I am currently working on a problem where I am graphing the trajectories of three masses (the ...
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Obtaining the star product from the Weyl quantisation of the product of two symbols

It can be shown (Groenewold 1946) that the Weyl quantisation of the product of two Weyl symbols is given by $$ [A(\textbf{r})B(\textbf{r})]_{w}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{4}}e^{i\...
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Canonical transformation in field theory

I am working with a condensed matter system in terms of charge and spin bosonic fields $\phi_c$ and $\phi_s$, respectively. I am trying to reach a diagonal Hamiltonian by diagonalizing the following ...
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How can non-chaotic curves fill a (hyper)torus and chaotic curves fill the entire energy hypersphere?

What I already know Before I ask my question, I would prefer to briefly explain what I already know, so that any gap in my understanding could be rectified. Note: I consider only bounded phase space ...
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Husimi $Q$-function of Infinite Square well

Eigen-Wavefunction of infinite square well is $$\psi(x)=\sqrt{2/l}\sin(n\pi x/l).$$ I want to write Husimi $Q$ function for infinite square well. General expression of Q function is $$Q=(1/2)\pi \...
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$ΔqΔp \geq \frac{\hbar}{2}$ while $h$ is the cell in phase space

I can show how: \begin{equation} ΔqΔp \geq \frac{\hbar}{2} \end{equation} In a general way. You just need to consider the commutation properties of 2 generic operators to do so. Both $p$ and $q$ ...
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