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trelek2
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How 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?
trelek2 said:How 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?
dreamspy said:...
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.
Bob_for_short said:dx=vdt so P =2 int(dt)/T = 1.
dreamspy said:Now we have:
x(t) = Acos(ωt) and v(t) = -Aωsin(ωt)
Bob_for_short said:Maybe Asin(ωt) = A[1-cos2(ωt)]1/2 = A[1-x2/A2]1/2, so you obtain v(x)?
A simple harmonic oscillator is a system that exhibits periodic motion, where the restoring force is directly proportional to the displacement from the equilibrium position. This type of motion can be seen in systems such as a mass-spring system or a pendulum.
The equation for a simple harmonic oscillator is F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position.
The probability density function for a simple harmonic oscillator describes the probability of finding the oscillator at a certain position or energy state. It is directly related to the wave function, which represents the amplitude of the oscillation at a given point in space and time.
The amplitude of a simple harmonic oscillator does not affect the overall shape of the probability density function, but it does affect the maximum probability. As the amplitude increases, the maximum probability shifts towards the outer regions of the oscillation, indicating a higher likelihood of finding the oscillator at these points.
Several factors can affect the probability density function of a simple harmonic oscillator, including the amplitude, frequency, and energy of the oscillator, as well as any external forces or damping present in the system. Additionally, the shape and boundary conditions of the system can also have an impact on the probability density function.