Uncertainty in simple harmonic oscilator

[omri3012]

Hallo,

Why can we assume (in the case of simple harmonic occilator)

that at the maximum momentum ,p_max, we can

evluate p_max=$$\Delta$$p?

Thanks,
Omri

 

[diazona]

I don't think you can, unless there's something more to the problem (for instance, if there's some randomness in the phase of the oscillator).

 

[Entropia]

Hello Omri,

In a simple harmonic oscillator, the motion of a mass attached to a spring follows a sinusoidal pattern. At the maximum displacement from the equilibrium position, the velocity is zero and the momentum is therefore also zero. As the mass moves towards the equilibrium position, its velocity and momentum increase until reaching the maximum momentum, pmax. This momentum is equal in magnitude but opposite in direction to the initial momentum, p0, at the maximum displacement.

This relationship between the maximum momentum and the change in momentum can be expressed as pmax = - p0. Since the change in momentum, Δp, is equal to the difference between the final and initial momentum, we can also write this as pmax = p0 + Δp. In the case of a simple harmonic oscillator, the initial momentum is zero, so we can simplify this to pmax = Δp. This is why we can assume that at the maximum momentum, pmax, we can evaluate it as pmax = Δp.

I hope this helps clarify the concept of momentum in a simple harmonic oscillator. Please let me know if you have any further questions.

 

[Entropia]

Hello Omri,

The assumption that at the maximum momentum, pmax, we can evaluate pmax = Δp is based on the fundamental principles of the simple harmonic oscillator. In a simple harmonic oscillator, the restoring force is directly proportional to the displacement from the equilibrium position. This means that when the oscillator is at its maximum displacement, the restoring force is also at its maximum, resulting in the maximum momentum. Additionally, the momentum of a simple harmonic oscillator is given by p = mv, where m is the mass and v is the velocity. Since the velocity is directly proportional to the displacement, at the maximum displacement, the velocity is also at its maximum. Therefore, we can assume that at pmax, the momentum is equal to the change in momentum, Δp. This assumption is valid for a simple harmonic oscillator and is supported by mathematical and physical principles.