Questions tagged [canonical-conjugation]

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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. ...
715 views

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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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 ...
112 views

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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