Hamiltonian problem

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Offline Pmb

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Hamiltonian problem
« on: 30/03/2013 23:20:38 »
Hi folks,

I'm trying to solidify my QM so I'm studying Liboff's QM text 4th Ed. On page 23 he asks to show that A = x - pt/m satisfies the equation (@ = partial sign) @A/@t = -{A,H} where {A,H} is the Poisson bracket of A and H. Here H is the Hamiltonian. I tried simple substitution using Hamilton's equations but it didn't seem to work. :(

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Offline imatfaal

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Re: Hamiltonian problem
« Reply #1 on: 31/03/2013 00:45:36 »
Pete

Beyond my ken - but it rings bells with integrable dynamical systems, the liouville equation and that the poisson bracket of a function F(p,q) with p is somehow equivalent to the partial derivation with respect to q.   

[tex]\frac {\partial A}{\partial t} = \left\{ A, H\right\}[/tex]


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Offline syhprum

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Re: Hamiltonian problem
« Reply #2 on: 31/03/2013 08:39:01 »
Have you tried looking in Wolfram Mathematica ? it has many references to Hamilton's
syhprum

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Offline JP

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Re: Hamiltonian problem
« Reply #3 on: 31/03/2013 16:05:53 »
Don't have time to type out the detail, but I think...

Assuming the Hamiltonian is p2/(2m),
Poisson bracket

Partial derivative of A wrt time

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Offline lightarrow

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Re: Hamiltonian problem
« Reply #4 on: 31/03/2013 18:29:48 »
Hi folks,

I'm trying to solidify my QM so I'm studying Liboff's QM text 4th Ed. On page 23 he asks to show that A = x - pt/m satisfies the equation (@ = partial sign) @A/@t = -{A,H} where {A,H} is the Poisson bracket of A and H. Here H is the Hamiltonian. I tried simple substitution using Hamilton's equations but it didn't seem to work. :(
But you or the book forgot to say that dA/dt = 0.

A = x - pt/m  -->  @A/@t = -p/m

{A,H} = @A/@x @H/@p - @A/@p @H/@x = @H/@p + t/m @H/@x

Now you use Hamilton's equations:
dx/dt = @H/@p
dp/dt = - @H/@x

and {A,H} becomes: (dx/dt - t/m dp/dt)

dA/dt = (total derivative of A) = (dx/dt - t/m dp/dt) - p/m = {A,H} - p/m 

-->  {A,H} = dA/dt + p/m = dA/dt - @A/@t  -->

 @A/@t = dA/dt - {A,H} = - {A,H}
« Last Edit: 31/03/2013 18:36:16 by lightarrow »

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Offline imatfaal

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Re: Hamiltonian problem
« Reply #5 on: 31/03/2013 18:46:20 »
+1 Lightarrow.  if A is conserved dA/dt =0 - and even I can follow the rest...
Thereís no sense in being precise when you donít even know what youíre talking about.  John Von Neumann

At the surface, we may appear as intellects, helpful people, friendly staff or protectors of the interwebs. Deep down inside, we're all trolls. CaptainPanic @ sf.n

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Offline Pmb

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Re: Hamiltonian problem
« Reply #6 on: 31/03/2013 21:32:22 »
Quote from: lightarrow
But you or the book forgot to say that dA/dt = 0
Neither I nor the book forgot to say that since that's what is to be shown, i.e. the problem reads
Quote
For a free particle moving in one dimension, show that

A = x - pt/m

satisfies the equation

@A/@t = -{A, H}

so that it is a constant of the motion.
I see that I didn't state the problem completely. The particle is free and therefore dp/dt = 0.

In the problem before that I showed that

dA/dt = @A/@t + {A, H}

Therefore the problem tells us to show that A = x - pt/m satisfies @A/@t = -{A, H} so that dA/dt = 0 and therefore A = constant. Your response is not the answer because you assumed that which was to be proven.

I know how to solve it now

A = x - pt/m     

@A/@t = -p/m = -v = -dx/dt

@A/@p = t/m

@A/@x = 1

Hamilton's equations

@H/@x = -dp/dt = 0

@H/@p = dx/dt

{A, H} = (@A/@x) (@H/@p) - (@H/@x) (@A/@p)

{A, H} = dx/dt - (-dp/dt) (t/m) = dx/dt = -@A/@t

i.e.

@A/@t = -{A, H}   QED!!
« Last Edit: 01/04/2013 00:29:38 by Pmb »

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Offline lightarrow

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Re: Hamiltonian problem
« Reply #7 on: 01/04/2013 18:25:58 »
+1 Lightarrow.  if A is conserved dA/dt =0 - and even I can follow the rest...
Thanks!

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Offline lightarrow

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Re: Hamiltonian problem
« Reply #8 on: 01/04/2013 18:28:26 »
Your response is not the answer because you assumed that which was to be proven.
...
I know how to solve it now
...
It's your way of saying "thank you for the hint!" ?  [:)]

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Offline Pmb

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Re: Hamiltonian problem
« Reply #9 on: 01/04/2013 21:57:01 »
Your response is not the answer because you assumed that which was to be proven.
...
I know how to solve it now
...
It's your way of saying "thank you for the hint!" ?  [:)]
Not really. I couldn't figure it out from what you posted, especially what you did with the p/m term. A friend of mine e-mailed  me and reminded me it was a free particle and thus dp/dt =0. It was a cinch from there. Not your fault though since I left that part out. But that you very much for the effort. :)

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Offline lightarrow

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Re: Hamiltonian problem
« Reply #10 on: 03/04/2013 10:34:27 »
A friend of mine e-mailed  me and reminded me it was a free particle and thus dp/dt =0.
And the text of the exercise don't remind it? Weird.

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Offline Pmb

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Re: Hamiltonian problem
« Reply #11 on: 04/04/2013 16:18:36 »
A friend of mine e-mailed  me and reminded me it was a free particle and thus dp/dt =0.
And the text of the exercise don't remind it? Weird.
It's known as a brain fart. Give me a break. I don't screw up all that often. :)  Plus and I live with terrible constant nagging pain that distracts me which messes up my concentration big time.

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Offline lightarrow

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Re: Hamiltonian problem
« Reply #12 on: 05/04/2013 08:38:43 »
It's known as a brain fart. Give me a break. I don't screw up all that often. :)  Plus and I live with terrible constant nagging pain that distracts me which messes up my concentration big time.
I didn't mean to make a critic to you, was just curious.

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Offline Pmb

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Re: Hamiltonian problem
« Reply #13 on: 05/04/2013 10:56:24 »
It's known as a brain fart. Give me a break. I don't screw up all that often. :)  Plus and I live with terrible constant nagging pain that distracts me which messes up my concentration big time.
I didn't mean to make a critic to you, was just curious.
Oh. Sorry. My mistake. I was embarrased for missing something simple.