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Harmonic oscilator in analytical mechanics(with the use of creation/annihilation ops)

  1. Nov 29, 2009 #1
    1. The problem statement, all variables and given/known data
    The Hamiltonian for the one-dimensional harmonic oscillator is given by:
    [tex] H = \frac{p^2}{2m}+ \frac{mw^2q^2}{2} [/tex]


    2. Relevant equations

    (a) Express H in terms of the following coordinates:

    [tex] a = \sqrt{\frac{mw}{2}} (q+i\frac{p}{mw}) [/tex]
    [tex] a^* = \sqrt{\frac{mw}{2}} (q-i\frac{p}{mw}) [/tex]

    (b) Calculate the following Poisson Brackets: {a*,H} {a,H}, {a, a*}
    (c) Write and solve the equations of motion for a and a.

    3. The attempt at a solution

    a. simple algebra :
    [tex] H = waa^* [/tex]

    b. again, algebra :
    [tex] \{a^*,H\} = iwa^*, \{a,H\} = -iwa, \{a,a^*\} = -i [/tex]

    c.

    this is where I'm having trouble. I don't really get the question.. what's the conjugate momentum? what's the coordinates? where do I start from? :)

    Thanks
     
    Last edited: Nov 29, 2009
  2. jcsd
  3. Nov 29, 2009 #2
    Re: Harmonic oscilator in analytical mechanics(with the use of creation/annihilation

    Perhaps you should start with, 'what does it mean "equations of motion"?' This means, in Hamiltonian dynamics, that you need to find the time-derivative of a and a-star.
     
  4. Nov 29, 2009 #3
    Re: Harmonic oscilator in analytical mechanics(with the use of creation/annihilation

    Also, I think part a is wrong. You may want to double check your algebra.
     
  5. Nov 30, 2009 #4
    Re: Harmonic oscilator in analytical mechanics(with the use of creation/annihilation

    I rechecked part a again, and also solved it using another method, got the same answer. I also think It's right since it resembles the quantum hamiltonian, without [tex] \hbar [/tex] and [tex] \frac{1}{2} [/tex], which makes sense.

    For the third part, I used [tex] \dot{a} = \{a,h\} [/tex] and solved it.
    Before I did that simple thing, I expressed q and p using a, a*, wrote an expression for [tex] \dot{a}, \dot{a^*} [/tex] and then used the hamiltonian equations of motion to express [tex] \dot{p}, \dot{q} [/tex] using a, a*, and got the same result :)

    Thanks for your help :smile:
     
  6. Nov 30, 2009 #5
    Re: Harmonic oscilator in analytical mechanics(with the use of creation/annihilation

    The way I see it, it pretty much is the quantum harmonic oscillator with [itex]\hbar=1[/itex] units:

    [tex]
    q^2=\frac{1}{2m\omega}\left(a+a^*\right)^2
    [/tex]

    [tex]
    p^2=-\frac{m\omega^2}{4}\left(a-a^*\right)^2
    [/tex]

    so that

    [tex]
    \begin{array}{ll}
    H&=\frac{p^2}{2m}+\frac{m\omega^2q^2}{2} \\
    &=-\frac{\omega}{4}\left(aa-aa^*-a^*a+a^a^*\right)+\frac{\omega}{4}\left(aa+aa^*+a^*a+a^*a^*\right) \\
    &=\frac{\omega}{4}\left(2aa^*+2a^*a\right) \\
    &=\omega\left(aa^*+\frac{1}{2}\right)
    \end{array}
    [/tex]

    It looks, though, that the factor of [itex]\frac{1}{2}[/itex] isn't necessary for parts (b) and (c), just part (a)--this is because you can solve the Poisson brackets in terms of [itex]q[/itex] and [itex]p[/itex], in which case you can use the Hamiltonian you began with.
     
    Last edited: Nov 30, 2009
  7. Nov 30, 2009 #6
    Re: Harmonic oscilator in analytical mechanics(with the use of creation/annihilation

    I think you are treating a, a* as operators (used commuting relation in the last part, if I'm not mistaken), while here they are simply coordinates, and aa* = a*a.
    Please note that you expressed [tex] p^2, q^2 [/tex] in a wrong way but then corrected it in H.

    Also, my algebra is

    [tex] waa^* = \frac{mw^2}{2} (q^2 + \frac{p^2}{(mw)^2}) = \frac{p^2}{2m} + \frac{mw^2q^2}{2} = H [/tex]

    I also did this by expressing q and p using a, a* and got the same result.
     
  8. Nov 30, 2009 #7
    Re: Harmonic oscilator in analytical mechanics(with the use of creation/annihilation

    Ha, that I am. I have been stuck on my QM for the last several days before my final on Wednesday--of course it doesn't help that your title says "creation/annihilation ops". Your answer is right then.


    Right, there should have been a squared term on the parenthesis--just fixed those too. Glad I could help!!
     
  9. Nov 30, 2009 #8
    Re: Harmonic oscilator in analytical mechanics(with the use of creation/annihilation

    If I remember correctly, the half in the quantum hamiltonian is the vacuum energy, and is a quantum effect, not classical. (This all may be wrong, don't take me up for it :) )
     
  10. Nov 30, 2009 #9
    Re: Harmonic oscilator in analytical mechanics(with the use of creation/annihilation

    Good luck on your exam!
     
  11. Nov 30, 2009 #10
    Re: Harmonic oscilator in analytical mechanics(with the use of creation/annihilation

    Thanks!
     
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