Harmonic Oscillator Analytical Mechanics: Creation/Annihilation Ops

[Loxias]

Homework Statement

The Hamiltonian for the one-dimensional harmonic oscillator is given by:
$$H = \frac{p^2}{2m}+ \frac{mw^2q^2}{2}$$

Homework 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.

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

 

[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!