# Velocity distribution

1. Feb 18, 2006

### 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.

2. Feb 18, 2006

### Tide

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

3. Feb 19, 2006

### rbj

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.

4. Feb 19, 2006

### 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)

5. Feb 19, 2006

### rbj

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 sorta way if the wave number is high enough (which is evidence of the correspondance principle).

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

Last edited: Feb 20, 2006
6. Feb 19, 2006

### Mulder

Thanks

Not something I remember explicitly seeing before.

7. Feb 20, 2006

### rbj

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