Probability is equal to the time spent in interval (x,x+dx) normed to total time it takes mass to run through entire available x space ( from -A to A, A is amplitude ). So you haveHow to find the probability density function of a simple harmonic oscillator? I know that for one normal node is should be a parabola but what is the formula and how do we derive it?
dx=vdt so P =2 int(dt)/T = 1....
Now if we integrate dP we get
P = int( 2/(T*v) dx)
But since I only have v(t), but not v(x), I'm not sure how to go about this.
Which is what we started with ( dp = 2 dt/T ), with normalization added: P(-infinity < x < +infinity) = 1dx=vdt so P =2 int(dt)/T = 1.
Maybe Asin(ωt) = A[1-cos^{2}(ωt)]^{1/2} = A[1-x^{2}/A^{2}]^{1/2}, so you obtain v(x)?Now we have:
x(t) = Acos(ωt) and v(t) = -Aωsin(ωt)
Thats seems to be about right, although I don't understand how you got there :)Maybe Asin(ωt) = A[1-cos^{2}(ωt)]^{1/2} = A[1-x^{2}/A^{2}]^{1/2}, so you obtain v(x)?