Obtaining Velocity Distribution P(v) for Simple Harmonic Motion

[Mulder]

Is it possible to analytically obtain a velocity distribution P(v) for a particle, say, undergoing simple harmonic motion v=sin(wt) (between max v' and min -v', say)

I'm not sure if this is obvious, I've not come across it before.

Cheers for any feedback.

 

[Tide]

A distribution function only makes sense when you have a lot of particles.

 

[rbj]

A distribution function only makes sense when you have a lot of particles.

or when you sample the particle velocity at random times. i remember seeing something like this for the simple harmonic oscillator in QM and comparing QM distribution to the Classical distribution.

 

[Mulder]

Ok, say a classical particle in a box oscillates with a sawtooth displacement with time - it either has velocity v_0 or -v_0, then I can write a velocity distribution like

P(v)=1/2(delta (v-v_0)+delta (v+v_0)

possible for any other kind of motion?

(I'll use latex one day)

 

[rbj]

Ok, say a classical particle in a box oscillates with a sawtooth displacement with time - it either has velocity v_0 or -v_0, then I can write a velocity distribution like

P(v)=1/2(delta (v-v_0)+delta (v+v_0)

possible for any other kind of motion?

sure. say it was a sinusoidal oscillation.

$$x(t) = A \mbox{sin}(\omega t + \theta)$$

and you sample its position at some random time. the p.d.f. of the position is

$$p_x(\alpha) = \frac{1}{\pi \sqrt{A^2 - \alpha^2}}$$ (for $|\alpha| < A$, zero otherwize)

independent of $\theta$.

we know what the velocity function is:

$$v_x(t) = x^{\prime}(t) = A \omega \mbox{cos}(\omega t + \theta) = A \omega \mbox{sin}(\omega t + \theta + \pi/2)$$

so the same can be applied to the velocity function (if sampled a random time):

$$p_v(\alpha) = \frac{1}{\pi \sqrt{(A \omega)^2 - \alpha^2}}$$ (for $|\alpha| < A \omega$, zero otherwize)

and the QM model of the harmonic oscillator will begin to look like that in an average sort of way if the wave number is high enough (which is evidence of the correspondance principle).

(I'll use latex one day)

it's useful for condoms. (pretty worthless for math.)

 

[Mulder]

Thanks


Not something I remember explicitly seeing before.

 

[rbj]

Thanks


Not something I remember explicitly seeing before.

quite all right. note that i had to fix the pdf functions a little.