# 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. ...
1 vote
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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 ...
1 vote
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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,...
1 vote
54 views

### 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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### 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 ...
591 views

### 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 ...
155 views

### 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 ...
1 vote
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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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