probability density function

Examples of probability density function in the following topics:

• Speed Distribution of Molecules

  • The most probable speed vp (at the peak of the curve) is less than the rms speed vrms.
  • Maxwell-Boltzmann distribution is a probability distribution.
  • where fv is the velocity probability density function.
  • (Derivation of the formula goes beyond the scope of introductory physics. ) The formula calculates the probability of finding a particle with velocity in the infinitesimal element [dvx, dvy, dvz] about velocity v = [vx, vy, vz] is:
  • Integration of the normal probability density function of the velocity, above, over the course (from 0 to $2\pi$) and path angle (from 0 to $\pi$), with substitution of the speed for the sum of the squares of the vector components, yields the following probability density function (known simply as the Maxwell distribution):

• Philosophical Implications

  • Even fundamental issues, such as Max Born's basic rules interpreting ψ*ψ as a probability density function took decades to be appreciated by society and many leading scientists.
  • However, the Copenhagen interpretation suggests a universe in which outcomes are not fully determined by prior circumstances but also by probability.
  • This is due to the quantum mechanical principle of wave function collapse.
  • That is, a wave function which is initially in a superposition of several different possible states appears to reduce to a single one of those states after interaction with an observer.

• The Wave Function

  • A wave function is a probability amplitude in quantum mechanics that describes the quantum state of a particle and how it behaves.
  • In quantum mechanics, a wave function is a probability amplitude describing the quantum state of a particle and how it behaves.
  • Although ψ is a complex number, |ψ|2 is a real number and corresponds to the probability density of finding a particle in a given place at a given time, if the particle's position is measured.
  • If these requirements are not met, it's not possible to interpret the wave function as a probability amplitude.
  • Relate the wave function with the probability density of finding a particle, commenting on the constraints the wave function must satisfy for this to make sense

• Problems

  • Let's suppose that the gas has a uniform density $\rho$ and consists of hydrogen with mass-fraction $X$ and helium with mass-fraction $Y$ and other stuff $Z$.You can assume that $Z/A=1/2$ is for the other stuff.What is the number density of electrons in the gas?
  • If you assume that the gas is spherical with radius $R$, what is the value of the Compton $y$-parameter as a function of $b$, the distance between the line of sight and the center of the cluster?
  • What is the synchrotron emission from a single electron passing through a magnetic field in terms of the energy density of the magnetic field and the Lorentz factor of the electron?
  • The number density of the electrons is $n_e$ and they fill aspherical region of radius $R$.What is the energy density of photons within the sphere, assuming that it is optically thin?
  • What is the inverse Compton emission from a single electron passing through a gas of photons field in terms of the energy density of the photons and the Lorentz factor of the electron?

• The H_2^+ Molecular Ion

  • where the semicolon in the $\chi_j$ function encourages us to think of ${\bf R}$ as a parameter.
  • After the electronic wavefuntion is calculated as a function of $R$, we can determine the proton wavefunction.
  • The second eqution in this section is generally to difficult to solve directly, so one generally picks a trial wavefunction and calculates the value of the energy for this function.
  • Figure 10.1 depicts the energy of the electronic configuration and Figure 10.2 shows the electron density for the two orbitals.
  • From the picture of the electron probability density we can see why this is the case.

• Special Properties of Water

  • At temperatures greater than 4ºC (40ºF) water expands with increasing temperature (its density decreases).
  • Upon freezing, the density of water decreases by about 9%.
  • The density of water as a function of temperature.
  • The maximum density at +4ºC is only 0.0075% greater than the density at 2ºC, and 0.012% greater than that at 0ºC.

• Problems

  • Calculate the ionized fraction of pure hydrogen as a function of the density for a fixed temperature.

• Problems

  • The sphere is held at uniform temperature, $T_0$, uniform density and constant mass $M_0$ during the collapse and has decreasing radius $R_0$.
  • What is the total luminosity of the sphere as a function of $M_0, R(t)$ and $T_0$while the sphere is optically thin?
  • What is the luminosity of the sphere as a function of time after it becomes optically thick in terms of $M_0, R(t)$ and $T_0$?
  • Draw a curve of the luminosity as a function of time.

• Variation of Pressure With Depth

  • The pressure exerted by a static liquid depends only on the depth, density of the liquid, and the acceleration due to gravity. gives the expression for pressure as a function of depth within an incompressible, static liquid as well as the derivation of this equation from the definition of pressure as a measure of energy per unit volume (ρ is the density of the gas, g is the acceleration due to gravity, and h is the depth within the liquid).
  • For any given liquid with constant density throughout, pressure increases with increasing depth.
  • As a result, pressure within a liquid is therefore a function of depth only, with the pressure increasing at a linear rate with respect to increasing depth.
  • In practical applications involving calculation of pressure as a function of depth, an important distinction must be made as to whether the absolute or relative pressure within a liquid is desired.
  • This equation gives the expression for pressure as a function of depth within an incompressible, static liquid as well as the derivation of this equation from the definition of pressure as a measure of energy per unit volume (ρ is the density of the gas, g is the acceleration due to gravity, and h is the depth within the liquid).

• Statistical Mechanics

  • The energy density ($u_\nu(\Omega)$)of the radiation field is simply the density of states times the mean energy per state and $c u_\nu(\Omega)=I_\nu$.
  • According to statistical mechanics the probability of a state of energy $E$ is proportional to $e^{-\beta E}$ where $\beta = 1/(k T)$.
  • If we use this value with the density of states we get the Wien law.
  • We have the value of energy density
  • The number density of photons can be determined in a similar way but the exponent in the integral is "2" instead of "3" yielding