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

[Loxias]

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


2. Relevant equations

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

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

(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 :
H = waa^*

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

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

[jdwood983]

(c) Write and solve the equations of motion for a and a. [my edit--I assume you mean a and a*?]

...
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

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.

[jdwood983]

Also, I think part a is wrong. You may want to double check your algebra.

[Loxias]

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 \hbar and \frac{1}{2} , which makes sense.

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

Thanks for your help :smile:

[jdwood983]

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 \hbar and \frac{1}{2} , which makes sense.


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


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



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


so that


\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}


It looks, though, that the factor of \frac{1}{2} isn't necessary for parts (b) and (c), just part (a)--this is because you can solve the Poisson brackets in terms of q and p, in which case you can use the Hamiltonian you began with.

[Loxias]

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 p^2, q^2 in a wrong way but then corrected it in H.

Also, my algebra is

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

I also did this by expressing q and p using a, a* and got the same result.

[jdwood983]

I think you are treating a, a* as operators (used commuting relation in the last part, if I'm not mistaken),

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.



Please note that you expressed p^2, q^2 in a wrong way but then corrected it in H.


Right, there should have been a squared term on the parenthesis--just fixed those too. Glad I could help!!

[Loxias]

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 :) )

[Loxias]

Good luck on your exam!

[jdwood983]

Good luck on your exam!

Thanks!