Probability of Displacement for Linear Harmonic Oscillators

[Derivator]

Homework Statement

Assume, you have an ensemble of linear harmonic oscilators, all having the same frequency $$\omega$$ and amplitude $$a$$:

$$x = a\cos(\omega t + \phi)$$​

.

The phase $$\phi$$ is uniformly distributed in the inteval $$[0,2\pi)$$. What ist the probability $$w(x)dx$$ to find the displacement of one oscillator in the interval $$[x,x+dx]$$? (for constant $$t$$)

Homework Equations

The Attempt at a Solution

I have looked in all my books and in the internet, but I absolutely have no idea how to start...

A little hint would be great!

(I don't want a solution from you, I just want to know, what formula/concept I have to look at, to have at least a chance to solve this exercise...)

--derivator

 

[mark726]

Dear derivator,

Thank you for your question. This is a great problem and I am happy to help you find a solution. The concept that you need to use in this problem is called probability distribution. In probability theory, a probability distribution is a function that describes the likelihood of obtaining a certain outcome in a random experiment. In this case, we have a random experiment where we are measuring the displacement of an oscillator in the given interval.

To find the probability w(x)dx, we need to use the probability distribution function for a uniform distribution. This is given by:

f(x) = 1/(b-a), for a <= x <= b

Where a and b are the minimum and maximum values of the interval. In this case, a = 0 and b = 2π. Therefore, the probability distribution function for our problem is:

f(x) = 1/(2π), for 0 <= x <= 2π

Now, to find the probability w(x)dx, we need to integrate this function over the interval [x,x+dx]. This can be done using the following formula:

w(x)dx = ∫f(x)dx = ∫(1/(2π))dx = 1/(2π) * dx

Therefore, the probability w(x)dx is simply 1/(2π) times the width of the interval, dx. This means that the probability of finding the displacement of one oscillator in the interval [x,x+dx] is the same for all values of x within the interval, and is given by 1/(2π)dx.

I hope this helps you understand the concept and find a solution to the problem. Good luck!