# Harmonic Oscillator Analytical Mechanics: Creation/Annihilation Ops

• Loxias
In summary, the Hamiltonian for the one-dimensional harmonic oscillator is given by: H = \frac{p^2}{2m}+ \frac{mw^2q^2}{2}
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

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Loxias said:
(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.

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

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 :)

Loxias said:
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.

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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.

Loxias said:
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.
Loxias said:
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!

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 said:

Thanks!

## 1. What is a harmonic oscillator in analytical mechanics?

A harmonic oscillator in analytical mechanics is a system that follows the laws of harmonic motion, where the restoring force is proportional to the displacement from equilibrium. It can be described using the creation and annihilation operators, which are mathematical tools used to describe the quantum states of a system.

## 2. How do creation and annihilation operators work?

Creation and annihilation operators are mathematical operators that act on quantum states of a system. The creation operator adds energy to the system, while the annihilation operator removes energy. Together, they help describe the energy levels and transitions of a harmonic oscillator.

## 3. What is the significance of creation and annihilation operators in harmonic oscillators?

Creation and annihilation operators are significant because they allow us to easily describe the energy levels and transitions of a harmonic oscillator. They also help us understand the quantum mechanical properties of the system, such as its ground state and excited states.

## 4. Can creation and annihilation operators be used for other systems besides harmonic oscillators?

Yes, creation and annihilation operators are not limited to just harmonic oscillators. They can also be used to describe other systems in quantum mechanics, such as the quantum harmonic oscillator, the hydrogen atom, and the simple harmonic oscillator.

## 5. How does the use of creation and annihilation operators simplify the analysis of harmonic oscillators?

The use of creation and annihilation operators simplifies the analysis of harmonic oscillators by providing a concise and elegant mathematical representation of the system. It allows us to easily calculate the energy levels and transitions of the system without having to use complicated equations or diagrams.

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