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Loxias

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## Homework Statement

The Hamiltonian for the one-dimensional harmonic oscillator is given by:

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

## Homework 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.

## 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

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