Canonical Variables: What Are They?

[peeyush_ali]

what are canonical variables..??
I'm a normal +2 student finished my intermediate college ..I just don't know about quantum physics..i understood that to understand better quantum physics i need to know about "canonical variables" ..

in hydrogen atom , consider 1st shell (ie n=1) the electron revolving around the nucleus have some angular momentum..but according to quantum theory, for n=1 , azimuthal quantum number or angular quantum number "l"=0 .. so is the electron revolving around the nucleus in a H atom doesn't have any angular momentum..?
what are these quantum numbers..? I really don't have a better understanding about them..I have just finished my classical physics and am not able to "take " or "digest" quantum theory any better...I would feel really ecstatic if anyone helps me out with this "quantum theory"...

 

[Andy Resnick]

"canonical" variables, in the sense that I am used to (Hamiltonian mechanics), essentially mean generalized variables. For example, instead of defining the position of a particle in terms of (x,y,z) coordinates, we can use the canonical position q_i, where i = 1,2,3. In Hamiltonian formulations of mechanics, the (canonical) position and (canonical) momentum variables are 'conjugate' to each other, where 'conjugate' implies specific relationships between the two (Poisson brackets, or Dirac brackets)

 

[peeyush_ali]

what is "hamiltonian mechanics"? can you give me a brief but clear cut idea about it?

 

[Andy Resnick]

The Hamiltonian of a system is a way to express the total energy in terms of coordinates and momenta. Many idealized dynamic systems can be expressed in terms of a Hamiltonian, and analysis of the equation of motion is reasonably straighforward. Using a Hamiltonian allows analysis of systems in terms of potentials rather than forces, and so is more general and more broadly applicable.

 

[Peeter]

There's a very readable treatment to be found here:

http://www.damtp.cam.ac.uk/user/tong/dynamics.htm

 

[adkinje]

Lenny Susskind from standford university has several lectures on youtube explaining both lagrangian and hamiltonian mechanics. They are just alternative formulations of the laws of motion and classical mechanics in general. Just find stanford's page on youtube and then find the playlist (modern physics - classical mechanics.)

They are really good lectures.

 

[Bob_for_short]

in hydrogen atom , consider 1st shell (ie n=1) the electron revolving around the nucleus have some angular momentum..but according to quantum theory, for n=1 , azimuthal quantum number or angular quantum number "l"=0 .. so is the electron revolving around the nucleus in a H atom doesn't have any angular momentum..?
what are these quantum numbers..? I really don't have a better understanding about them..I have just finished my classical physics and am not able to "take " or "digest" quantum theory any better...I would feel really ecstatic if anyone helps me out with this "quantum theory"...

Quantum mechanics is well understood as a wave mechanics. Standing waves in a limited medium have discrete frequencies. In atomic example you mention (l=0) the wave is purely radial. The negative charge cloud does not revolve but oscillates radially.

Bob_for_short.

 

[andresordonez]

what are canonical variables..??
I'm a normal +2 student finished my intermediate college ..I just don't know about quantum physics..i understood that to understand better quantum physics i need to know about "canonical variables" ..

in hydrogen atom , consider 1st shell (ie n=1) the electron revolving around the nucleus have some angular momentum..but according to quantum theory, for n=1 , azimuthal quantum number or angular quantum number "l"=0 .. so is the electron revolving around the nucleus in a H atom doesn't have any angular momentum..?
what are these quantum numbers..? I really don't have a better understanding about them..I have just finished my classical physics and am not able to "take " or "digest" quantum theory any better...I would feel really ecstatic if anyone helps me out with this "quantum theory"...

Ok, I'll try to make this clear from the beginning cause it seems to me that everyone has been somewhat confusing about this.

There are two courses commonly known as "classical mechanics". The first of these courses is an introduction to Newton laws, energy, etc and a book like Serway's is normally used. The second is a more advanced course where you learn about generalized variables, Lagrangian, Hamiltonian, etc and a book like Goldstein's is used. I guess you have only taken the first course.

I just finished the first course in quantum mechanics, and used the book by Cohen-Tannoudji (very good book, not that I know others though). I don't think it is completely necessary to take the second course of Classical Mechanics before taking the first Quantum Mechanics course, but it's a good idea to do it.

Anyway, I think you can do fine without the second course of Classical Mechanics (for now), but you must choose the right book to learn Quantum Mechanics, I would recommend the book I used, but as I said before, I don't know other books, so maybe you should ask around what book is the best for you.

On the other hand you can check out a Modern Physics book, I used Serway's Modern Physics. I don't think you're suppossed to take the second course in Classical Mechanics before reading that book so it would be a good idea to start there.