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5

For a 2-dimensional phase space, they are the same. More generally, for a $2n$-dimensional symplectic manifold $(M;\omega)$ with symplectic two-form $$\tag{1} \omega~=~\frac{1}{2} dz^I ~\omega_{IJ} \wedge dz^J, \qquad \omega_{IJ}~=~-\omega_{JI},$$ the Poisson bracket is given by $$\tag{2} \{f,g\}_{PB}~=~\frac{\partial f}{\partial z^I}\pi^{IJ} ... 0 I) As a purist, I disprove of the common praxis to call the implication$$ \tag{1} \{Q,H\}+\frac{\partial Q}{\partial t}~=~0 \qquad\Rightarrow\qquad \frac{dQ}{dt}~\approx~0.$$for a 'Hamiltonian version of Noether's theorem', as explained in my Phys.SE answer here. Moreover, the implication (1) is not equivalent to the full Noether's theorem for various ... 1 Comments to the question (v3): Eqs. (1) are part of the CCR for a scalar field, such as, e.g., a real or complex Klein-Gordon field, a Schrödinger field, etc. Eq. (2) refers to the Schrödinger field, which is a complex field, see e.g. this Phys.SE post. A real Schroedinger field does naively not make sense since e.g. the expected kinetic term \propto ... 2 The point is that the equation of motion of the fields is different if referring to temporal derivatives. In relativistic field theory, it is a second-order one and you need two initial conditions i.e. \pi and \pi to solve it. Quantizing, and interpreting the Fourier coefficients of the initial conditions as creation and annihilation operators, ... 1 So the Hamiltonian is H = \frac12 m v^2 + \frac12 m \omega^2 x^2and therefore we can define u = v/\omega to find that the circle swept out (of radius x = a) has ux-area \pi a^2 or px-area of \pi~m~\omega~a^2. Presumably you mean that this area is 2\pi E/\omega, independent of mass at constant energy and frequency. Other combinations will lead ... 0 It seems like you've got lost in the subject. To clarify some facts: The action for General Relativity (Einstein-Hilbert action) is, as usual, an integral of the Lagrangian density over spacetime:$$ S[g] = \frac{1}{16 \pi G} \int d^4 x \sqrt{-g} \cdot R, $$where \sqrt{-g} is the square root of the determinant of the metric tensor and R is the Ricci ... 0 Here is how I understand it: let's say you have some coordinates p_i and some momenta q_i. You want to find transformations$$Q_i\equiv Q_i(q_i,\,...,\,q_n,\,p_i,...,\,p_n,\,t)\qquad P_i\equiv P_i(q_i,\,...,\,q_n,\,p_i,...,\,p_n,\,t) $$These variables p_i, q_i, Q_i, and P_i must satisfy the Hamilton's equations of motion: ... 1 This answer outlines how the defining matrix representation of the symplectic group Sp(2m,R) is ray optics, whilst the infinite-dimensional unitary rep of Sp(2m,R) carried on the space of wavefunctions is diffractive optics in the Fresnel approximation. The outline is for Sp(2,R) (cylindrical lenses) but the generalization to Sp(2m,R) is reasonably ... 1 The Weyl system \exp\left[\frac{i}{\hbar} Q \hat{p}\right] and \exp\left[\frac{i}{\hbar}P\hat{q}\right] comprise two "presentation" elements of the Heisenberg group. To the extent \hat{p} is a derivative with respect to position q, Q is the shift amount q in any function of it is translated by the action of \exp\left[\frac{i}{\hbar} Q ... 18 Well, \{f, \cdot \}, similarly to \{H,\cdot\}, computes the derivative of the argument \cdot with respect to the action of the one-parameter group of canonical transformations generated by f (see the note below for the complete definition)$$\phi_a^{(f)} : F \to F\:,\quad a \in \mathbb R\:,$$satisfying$$\phi_a^{(f)} \circ \phi_b^{(f)}= ...

2

Yes it does. In fact, it is one of the( if not the) most important conclusions of Quantum mechanics. If {f,g}= 0 it means that the variables have simultaneous eigenvalues i.e. you can measure both of them on same instant of time. but {f,g} can be non-zero which leads to the theoretical conchusion that the eigenstates of f and g are not simutaneously ...

1

The energy $E$ of an oscillator is given by $$E=\frac{p^2}{2m}+\frac12m\omega_1^2x^2$$ This defines an ellipse in phase space! So now, when $E=E_1$ everything within the ellipse defined by $E_1$ will have energy less than $E_1$. To proceed with finding the limits of of integration, we consider the cases when the particles' have all kinetic or all ...

3

Let us define: $$\hat L=i\sum ^N_{j=1}\bigg(\frac{{p}_j}{m_j}\frac{\partial }{\partial q^j}+\vec {F}_j(\boldsymbol q )\frac{\partial }{\partial p_j}\bigg)$$The Liouville operator can be expanded in terms of components and a basis like any vector field. Let $\xi =(q^1,\dots ,q^n,p^1,\dots p^n)$ be a phase space vector. ...

5

OP's question (v1) is essentially asking Does the operator identity $$e^{\frac{it}{\hbar}[\hat{H},~\cdot~]}\hat{A}~ =~ e^{i\hat{H}t/\hbar}\hat{A}e^{-i\hat{H}t/\hbar} \tag{1}$$ have an analog using functions/symbols $H$ and $A$ rather than operators $\hat{H}$ and $\hat{A}$, respectively? The answer is: Yes, in terms of the Groenewold-Moyal star ...

1

I am not too sure of the homework policy on this forum so I won't answer your question directly, but I hope this helps you :) Starting from the Lagrange density $\mathcal L$; $$\mathcal L=\frac 12 \rho _0\dot \eta^2 +P_0\nabla \cdot \eta -\frac 12 \gamma P_0(\nabla \eta )^2$$ The equation of motion for the $\eta$ field is given ...

2

In local coordinates the canonical transformation to action angle coordinates $(q,p)\rightarrow (Q,P)$ can be related by, $$\boxed{P_i=\frac{1}{2\pi}\oint p_idq^i \ \ \ \ \ \text{and}\ \ \ \ \ Q^i=\frac{\partial }{\partial P_i}\int p_idq^i}$$ For Example: Consider the one dimensional harmonic oscillator with the following ...

1

$- \vec\nabla [\dfrac{1}{2m} (\vec p - \dfrac{q}{c}\vec A) \cdot (\vec p- \dfrac{q}{c}\vec A) + q\phi]=$ $- \vec\nabla [\dfrac{1}{2m} (\vec p - \dfrac{q}{c}\vec A) \cdot (\vec p- \dfrac{q}{c}\vec A)] -q\vec\nabla\phi=$ $\vec\nabla\left[ \frac{-\vec p\cdot\vec p +2 \dfrac{q}{c}\vec A \cdot \vec p- \dfrac{q}{c}\vec A\cdot\dfrac{q}{c}\vec A ... 3 The standard name for what you are seeking, is the Wigner transform, the inverse of the Weyl transform. (As the Weyl transform maps phase-space functions to operators.) for an arbitrary operator in any ordering, the Wigner transform follows a simple 1964 formula by Kubo, eqn (111) of Ref. 1, effectively the Fourier transform of the off-diagonal matrix ... 2 Indeed, the Bopp shift is a clumsy Lagrange translation operator transcription of the celebrated * product, a 4-variable integral transform, cf. eqns (12-15) in Ref. 1. There is an infinity of phase-space functions corresponding to differently ordered operators, as their ps and qs may be ordered in different ways with the intercalated * s enforcing ... 2 The geometry and topology of the relevant phase space is identical for both classical and quantum problems: it is the very same phase space. The scale of the former is the small$\hbar$limit of the latter. Extended WFs appear like δ-fctn spikes ("points") in the small$\hbar$limit, once the phase space-variables are suitably rescaled by$\sqrt{\hbar}\$. ...

2

Most graduate text books in Classical mechanics have (as their last two chapters) discussions of perturbation theory in classical mechanics. These (however) are not invariably readable, and will usually restrict the solution to problems that can be described by a Hamiltonian e.g. have no friction or dissipation. Goldstein, "Classical Mechanics" has such a ...

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