Questions tagged [canonical-conjugation]
The canonical-conjugation tag has no usage guidance.
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Are the canonical momentum and the corresponding generalized coordinates independent? [duplicate]
I know that for a lagranian $L=L(q_i, \dot{q_i},t)$ the canonical momentum is given by $p_i = \frac{\partial L}{\partial \dot{q_i}}$. The lagrangian being a function of the generalized coordinate, I ...
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Commutation relation between charge number and phase in superconductors
Consider an isolated Cooper pair box.
The charge number operator is
$$\hat{n} = \sum_{n=-\infty}^{+\infty} n |n\rangle \langle n|$$
where $|n \rangle$ is the state in which $n$ Cooper pairs have ...
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Relation between energy and time
I would like help in understanding something that has been causing me a lot of trouble recently: Why is energy always related to time in physics?
Examples include the 4-momentum, the energy-time ...
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Conjugate observables - can the commutation relations be generalised?
Conjugate variables are variables that are Fourier transforms of one another (that is they are Fourier transform duals) and consequently have an uncertainty relation existing between them. In quantum ...
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Why does the Pauli objection not disqualify the existence of the position operator?
According to the Pauli objection (see for example here or the answer to this question) there can be no time operator $\hat{T}$ canonically conjugate to the Hamiltonian $\hat{H}$ of a physical system ...
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Hamiltonian formalism (with symplectic form) for time-dependent Lagrangian
I have been working on some results that work for time-independent Lagrangians $L\Big(q,\dot{q}\Big)$ and return a Hamiltonian function
$$
H(q,\dot{q})=\dot{q}^i \frac{\partial L}{\partial \dot{q}^i}-...
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Canonical commutation relation on the spatial boundary of the hypersurface
Consider the equal time commutation relation of a field given on a $d$ dimensional spacelike hypersurface $\Sigma$ of a $d+1$ dimensional manifold given by
$$[\Pi(t, x), \Phi(t, x')] = i\hbar\delta^{(...
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Canonical transformations in Quantum Mechanics
Heisenberg's famous commutator of a pair of conjugated canonical variables is formulated for position and conjugated momentum
$$[q,p] = i\hbar.$$
Intuitively I would guess that it would also work ...
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Lagrangian and Hamiltonian Mechanics: Conjugate Momentum
I am a physics undergraduate student currently taking a classical mechanics course, and I am not able to understand what conjugate/canonical momentum is (physically). It is sometimes equal to the ...
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Commutation relation between pairs of conjugated variables
Suppose I have four operators $\hat{q}_1$, $\hat{p}_1$, $\hat{q}_2$ and $\hat{p}_2$ such that
$[\hat{q}_1, \hat{p}_1] = [\hat{q}_2, \hat{p}_2] = i$
$[\hat{q}_1, \hat{q}_2] = [\hat{p}_1, \hat{p}_2] = [...
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How to expand Electromagnetic fields in term of Laguerre–Gaussian (LG) beams?
I was studying canonical quantization for the electromagnetic fields. I know that we can expand our fields in other normal modes which one of them is the Laguerre–Gaussian (LG) wave set but in most ...
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Quantization of radiation with canonical conjugate variables
I am reading the Introduction to quantum optics book and I am a bit stuck at the quantization of electromagnetic field (around page 317).
The problem in this case is that we can't just express the ...
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Why is $[ \hat{A},\hat{B} ] \rightarrow i \hbar \text{{A, B}}$?
If we have two classical quantities $A$, $B$, and their corresponding quantum operators $\hat{A}$, $\hat{B}$, then their commutators and Poisson brackets are linked by
$$ [ \hat{A},\hat{B} ] \...
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Why is the canonical momentum operator for a charged particle not observable in an E&M background?
From the Wikipedia article on the momentum operator (https://en.wikipedia.org/wiki/Momentum_operator#Canonical_commutation_relation) the following operator (in position representation)
$$\hat{\bf{P}} ...
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Are velocity and momentum conjugate variables?
I have come across this statement in Wikipedia:
"In physical problems, Legendre transformation is used to convert functions of one quantity (such as velocity, pressure, or temperature) into ...
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Conjugate variables in gravity?
We know that in the traditional quantum mechanics the conjugate variables are position and momentum, but what is known about the elusive quantum gravity?
It came to my mind that if there is something ...
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How does the tautological one-form convert a velocity to a momentum?
The Wikipedia page on the "tautological one-form" $\theta$ says that it
is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus ...
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Does the Hamiltonian formulation of classical mechanics require an inner product on physical space?
The Hamiltonian formulation of classical mechanics is quite broad and flexible; one of the only nontrivial physical assumptions that need to be made is that the degrees of freedom are continuous ...
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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,
$$
\frac{\partial F\left(q_i, Q_i, t\right)}{\partial t}=-H\left(p_i, q_i, t\right)=-H\...
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Calculating conjugate momenta for a spin-2 field
Consider a symmetric spin-2 field $h_{\mu \nu}$. I have the following Lagrangian for this field:
$$\mathcal{L} = - \frac{1}{4}\left(\partial_{\lambda}h_{\mu \nu} \text{ } \partial_{\phi}h_{\alpha \...
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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}$$
for some $M=M(q,p,Q,P,t)$....
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Planck's constant and the Uncertainty principle
Why should the uncertainty in measurement of two conjugate variables, say $p$ and $x$, be of the order of Planck's constant or higher and not lower? What is so sacrosanct about $h$? I always think $h$ ...
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Is there no coordinate conjugate to $L_x$?
The conjugate momentum corresponding to $\phi$ (azimuthal angle in sp. polar coordinate) is $L_z$ (sometimes written $L_\phi$) which is frequently used in quantum mechanics. Why is there no coordinate ...
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Do we need to know first time derivatives of Klein-Gordon fields to make future predictions?
In the quantisation of the Klein Gordon field, because it has a second derivative, we need to know the values of the field and also the first derivative in order to predict future values of the field. ...
user84158
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Conjugate variables in thermodynamics vs. Hamiltonian mechanics
According to Wikipedia, the canonical coordinates $p, q$ of analytical mechanics form a conjugate variables' pair - not just a canonically conjugate one.
However, the "conjugate variables" I directly ...
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Time-dependent Canonical transformation
Suppose that the Hamiltonian of the mechanical system under analysis depends on two complex conjugate variables $a$ and $a^*$, so we have:
$$
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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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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