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Vicol

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- Thread starter Vicol
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In summary, the conversation discusses the possibility of finding an analogous density function in classical mechanics that shows the probability of finding a classical particle in a given interval of space. It is noted that for a simple harmonic oscillator, the probability distribution can be represented by the equation (1/2pi)(A^2-x^2)^-1/2. This function can also be used to calculate average energy, momentum, and position in classical mechanics, similar to how it is done in quantum mechanics. However, in order to accurately represent the probability, only half a period of oscillation should be considered.

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Vicol

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- #2

William Crawford

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In the case of a simple harmonic oscillator (SHO); your intuition is right, it is more likely to encounter it near its endpoints of oscillation.

If we choose our coordinate frame such that the origin is at the equilibrium point for the SHO and starts the oscillation at the displacement ##A##, then the equation of motion is \begin{align}x(t) = A\cos(\omega t)\end{align} where \begin{align}\omega = \sqrt{\frac{k}{m}}\end{align} and ##k## is the spring constant. The velocity is \begin{align}\frac{dx}{dt} = -A\omega\sin(\omega t) \end{align} and the period of oscillation is ##T = 2\pi/\omega##. Then the probability for encountering it during the time interval ##dt## is ##\vert dt/T\vert##, thus the probability for encountering in the interval ##dx## is \begin{align}\bigg\vert\frac{dt}{T}\bigg\vert &= \frac{dx}{2\pi A}\csc(\omega t) \\ &= \frac{dx}{2\pi}\frac{1}{\sqrt{A^2 - x^2}}.\end{align} You can therefore think of \begin{align}\frac{1}{2\pi}\big(A^2-x^2\big)^{-\frac{1}{2}}\end{align} as the probability distrubution for this SHO.**EDIT:** The probability for encountering it is of course twice that given in (6) as it passes through the interval ##dx## twice (one time in each direction) during a full period of oscillation. This can be fixed in the derivation by only considering half a period of oscillation instead of a full period.

If we choose our coordinate frame such that the origin is at the equilibrium point for the SHO and starts the oscillation at the displacement ##A##, then the equation of motion is \begin{align}x(t) = A\cos(\omega t)\end{align} where \begin{align}\omega = \sqrt{\frac{k}{m}}\end{align} and ##k## is the spring constant. The velocity is \begin{align}\frac{dx}{dt} = -A\omega\sin(\omega t) \end{align} and the period of oscillation is ##T = 2\pi/\omega##. Then the probability for encountering it during the time interval ##dt## is ##\vert dt/T\vert##, thus the probability for encountering in the interval ##dx## is \begin{align}\bigg\vert\frac{dt}{T}\bigg\vert &= \frac{dx}{2\pi A}\csc(\omega t) \\ &= \frac{dx}{2\pi}\frac{1}{\sqrt{A^2 - x^2}}.\end{align} You can therefore think of \begin{align}\frac{1}{2\pi}\big(A^2-x^2\big)^{-\frac{1}{2}}\end{align} as the probability distrubution for this SHO.

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A probability density function (PDF) in classical mechanics is a mathematical function that describes the probability of a particle being in a certain state or having a certain value for a given physical quantity. It is used to predict the likelihood of a particle occupying a particular position or having a particular momentum at a specific time.

A PDF in classical mechanics is similar to a wave function in quantum mechanics, as both describe the probability of a particle being in a certain state. However, in classical mechanics, the PDF is a function of position and momentum, while in quantum mechanics, the wave function is a function of position and time.

In classical mechanics, the PDF is calculated by taking the square of the absolute value of the wave function, which is a complex-valued function that describes the state of a particle. The PDF can also be calculated by integrating the squared wave function over all possible states.

A PDF is a function that describes the probability of a particle being in a certain state, while a probability distribution is a function that describes the probability of a particle having a certain value for a given physical quantity. In other words, a PDF describes the probability of a particle being in a specific state, while a probability distribution describes the probability of a particle having a specific value for a physical quantity.

A PDF is used in classical mechanics to predict the behavior of particles in a system. It can be used to calculate the probability of a particle being in a certain state, or to calculate the average value of a physical quantity for a group of particles. It is also used to determine the most probable state of a particle in a given system.

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