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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. :(


 

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.   

d3a36fead1da2c0f90f68d460e78e870.gif


 

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
 

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
 

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 »
 

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...
 

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 »
 

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!
 

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!" ?  :)
 

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. :)
 

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.
 

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.
 

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.
 

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.
 

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Re: Hamiltonian problem
« Reply #13 on: 05/04/2013 10:56:24 »

 

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