Questions tagged [canonical-conjugation]

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What does it mean for two variables to be canonically conjugate?

The word "canonical" has been used in many of my classes (canonical ensemble, canonical transformations, canonical conjugate variables) and I am not really sure what it means physically. ...
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How can you confirm that two variables are canonically conjugate using Poisson brackets?

Suppose you have two conjugate variables $q$ and $p$ that are canonically transformed into two other variables $Q$ and $P$. What needs to hold true for these variables in terms of Poisson brackets? I ...
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verify canonically conjugate variables by way of Poisson brackets

How do I verify using Poisson brackets that a set of variables are canonically conjugate? It's for a homework problem but I'm interested in a general approach/reference if possible. We are using ...
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Deriving the momentum operator in position basis from its commutation relation

Canonical conjugate operators in Quantum Mechanics typically (if not always?) satisfy a commutation relation of the form $$ [\hat{x},\hat{p}]\equiv\hat{x}\hat{p}-\hat{p}\hat{x}=i\hbar $$ Most famously,...
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Can a Hamiltonian include explicitly the derivative of the conjugate momentum, especially after a canonical transformation?

Can a Hamiltonian expression, say $H$ with $(q,p)$ as conjugate variable pair, include the total derivative of $p$ explicitly? That is, can we have $H=H(q,p,\dot{p})$? And, if so, what does it imply ...
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Does Canonical Variable have any physical meaning in Classical Mechanics?

Consider a canonical transformation from variable $(q,p) \rightarrow (Q,P)$ generated by the generating function $F(q,Q)=qQ$ so in this case $$p=\frac{\partial F}{\partial q}=Q\Rightarrow Q=p$$ and $$...
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Charge number and phase as canonical variables

In the derivation of the Hamiltonian of a Cooper Pair Box, it is stated that the junction phase difference variable $\delta$ and the charge number variable $N$ satisfy the canonical commutation ...
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Ladder operators vs. conjugate variables

In the book Introduction to Many-Body Physics by Piers Coleman, it states on page 12 that ... the particle field and its complex conjugate are conjugate variables. In other words, the particle ...
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Is every pair of conjugate variables associated with a Fourier transform?

For example, in quantum mechanics, the commutator of the position and momentum is $$[\hat{P_i} ;\hat{Q_j} ] =i\hbar\delta_{ij}\neq 0, i\neq j$$ I know that the position space representation of the ...
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Are electric and magnetic fields canonical conjugates?

When quantizing the electromagnetic fields in the context of quantum optics and quantum field theory, we often go for the vector potential $\mathbf{A}$ and its canonical momentum, which turns out to ...
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Canonically conjugate variable of rapidity [closed]

Does the rapidity $\theta \in \mathbb{R}$ have a canonically conjugate variable? More specifically, for some smooth function $f \in \mathcal{S}(\mathbb{R})$, by Plancherel's theorem we have (up to ...
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Using direct conditions to check if a gauge transformation is canonical

Gauge transformations are known to be canonical transformations, see here and here. Using direct conditions to check if they are, however, imposes a constraint on the gauge parameter $\Lambda(t,\...
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Thermodynamical conjugate variables

In thermodynamics the potentials are typically only a function of 2 variables, say $$U=U(S,V)$$ with entropy $S$ and volume $V$. I see that conjugate pairs $S,T$ or $p,V$ always have the unit of ...
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Transformation to canonically conjugate coordinates in Special Relativity

Just like we can Fourier transform a field from $$x^\mu = (ct, x,y,z) \rightarrow p^\mu = (E/c, p_x, p_y, p_z)$$ via a Fourier transform, for spherical coordinates can we Fourier transform in the same ...
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Can the Fock-Schwinger (radial) gauge condition be written as momentum space divergence?

The Lorenz Gauge can be written (in QED) as $\partial^{\mu}A_{\mu} = 0$ or equivalently as $p^{\mu}A_{\mu} = 0$. The Fock-Schwinger gauge is similar: $x^{\mu}A_{\mu} = 0$. Can it be written ...
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When can one find a canonically conjugate operator?

Suppose one is given a self-adjoint operator $A$ acting on an infinite dimensional separable Hilbert space $\mathcal{H}$. Under what conditions can one find an operator $B$ such that $[A,B] = i$? And ...
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Gauge freedom in Lagrangian corresponds to canonical transformation of Hamiltonian

I want to show that the gauge transformation $$L(q,\dot{q},t)\mapsto L^\prime(q,\dot{q},t):=L(q,\dot{q},t)+\frac{d}{dt}f(q, t)$$ corresponds to a canonical transformation of the Hamiltonian $H(p, q, ...
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Origin of conjugate variables in physical theories

Why do conjugate variables come in pairs? For example, in classical mechanics we have the generalized coordinates of position and momentum, and there is Jacobi's action-angle coordinates. Also, in the ...
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Four special types of canonical transformations

Let $(q,p) \mapsto (Q,P)$ be a diffeomorphism of phase space. Then this is a canonical transformation (CT) if $$p\dot{q}-H(q,p,t)=P\dot{Q}-K(Q,P,t) + \frac{dM}{dt}\tag{1}$$ for some $M=M(q,p,Q,P,t)$....
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Planck's constant and the Uncertainty principle

Why should the uncertainty in measurement of two conjugate variables, say $p$ and $x$, be of the order of Planck's constant or higher and not lower? What is so sacrosanct about $h$? I always think $h$ ...
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Is there no coordinate conjugate to $L_x$?

The conjugate momentum corresponding to $\phi$ (azimuthal angle in sp. polar coordinate) is $L_z$ (sometimes written $L_\phi$) which is frequently used in quantum mechanics. Why is there no coordinate ...
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Do we need to know first time derivatives of Klein-Gordon fields to make future predictions?

In the quantisation of the Klein Gordon field, because it has a second derivative, we need to know the values of the field and also the first derivative in order to predict future values of the field. ...
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4 votes
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Conjugate variables in thermodynamics vs. Hamiltonian mechanics

According to Wikipedia, the canonical coordinates $p, q$ of analytical mechanics form a conjugate variables' pair - not just a canonically conjugate one. However, the "conjugate variables" I directly ...
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Time-dependent Canonical transformation

Suppose that the Hamiltonian of the mechanical system under analysis depends on two complex conjugate variables $a$ and $a^*$, so we have: $$ H=H\left(a,a^*\right) $$ Hamilton equations read $$ ...
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In what sense are shot noise and photon pressure canonically-conjugate variables in the LIGO interferometer?

This week I saw a seminar by Kip Thorne and he mentioned that in the LIGO interferometer, the photon shot noise is actually canonically conjugate to the noise induced by photon pressure acting on the ...
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2 votes
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Why is entropy not a thermodynamic potential in the sense of Legendre transform?

We can formulate thermodynamic theory using entropy S as state function or equivalently using the internal energy U, and U is related with all other thermodynamic potentials via the Legendre transform,...
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How does one know if two variables are conjugate pairs?

First of all, I am having a hard time finding any good definition of what a conjugate pair actually is in terms of physical variables, and yet I have read a number of different things which use the ...
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Is it always possible to define/find conjugate variables? And if yes how one can find it?

My question is in the context of both classical and quantum mechancis and field theory. Generally, how can one define/find the (canonically) conjugate of some variable/operator/field? Examples ...
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Unitary translation in phase space coordinate

If we suppose that we can translate one point to another point in phase space $(x,p)$ through the following operators, $$T(\Delta x) = \exp(-i p~\Delta x ) $$ and $$T(\Delta p) = \exp(-i x~\Delta p ) ,...
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Key Assumption to derive the Uncertainty Principle --- Canonical Conjugate Operators [closed]

From this related question, Rigorous Mathematical Proof of the Uncertainty Principle from First Principles The key assumption to derive the uncertainty principle seems to be the relationship between ...
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Placement of indices in canonical commutation relations of coordinates and conjugate momenta as well as fields and conjugate momenta

The canonical commutation relations between generalised coordinates $q_a$ and their conjugate momenta $p^a$ are given by $[q_a,q_b]=[p^a,p^b]=0$ $[q_a,p^b]=i\delta^b_a$. Furthermore, the canonical ...
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What is the momentum canonically conjugate to spin in QM?

In Kopec and Usadel's Phys. Rev. Lett. 78.1988, a spin glass Hamiltonian is introduced in the form: $$ H = \frac{\Delta}{2}\sum_i \Pi^2_i - \sum_{i<j}J_{ij}\sigma_i \sigma_j, $$ where the ...
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6 votes
1 answer
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Non-hermiticity of Dirac Lagrangian: null momentum?

The usual Dirac Lagrangian is $L(\psi,\bar\psi)=\bar\psi(i\not\partial-m)\psi$. The canonical momenta are $$ \pi=\frac{\partial L}{\partial \psi_{,0}}=i\psi^\dagger \\ \bar \pi=\frac{\partial L}{\...
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3 votes
1 answer
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What is the difference between the momentum in the Fourier transform of a scalar field and the conjugate momentum of the field?

What is the difference between the momentum $p$ in $e^{i\mathbf{p}\cdot{\mathbf{x}}}$ in the Fourier transform of a scalar field and the corresponding conjugate momenta $\pi(x)$ of the scalar field?
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5 votes
2 answers
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Energy and momentum as partial derivatives of on-shell action in field theory

According to L&L, if we fix the initial position of a particle at a given time and consider the on-shell action as a function of the final coordinates and time, $S(q_1, \ldots, q_n, t)$, then... $...
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4 votes
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Lagrangian with vanishing conjugate momentum, independent variables

Given a Lagrangian density $\mathcal L(\phi_r,\partial_\mu\phi_r,\phi_n,\partial_\mu\phi_n)$, for which we find out that for some $\phi_n$ its conjugate momentum vanishes: $$\pi_n=\frac{\partial\...
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Why is momentum (instead of something else) the canonical conjugate of position?

Why did nature decide to make conjugate of position to be momentum? Since energy and position do not commute, why not energy? What determines the pairing of time with energy and momentum with position?...
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9 answers
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Why do quantum physical properties come in pairs?

Why do quantum physical properties come in pairs, governed by the uncertainty principle (that is, position and momentum?) Why not in groups of three, four, etc.?
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