Angular probability density function

It is useful to have a pictorial representation of these rotational states and the angular probability density function, offers some... 

The angular probability density function, P B, provides the basis for a pictorial description of the rotational states of dihydrogen. The function is a surface represented in polar coordinates. Any point on the surface can be joined to the origin by a line. The 0, (f) coordinates of the line correspond to the orientation of the dihydrogen molecular axis and the probability of any particular orientation occurring is given by the... 

In the fixed axis rotation model of dielectric relaxation of polar molecules a typical member of the assembly is a rigid dipole of moment p rotating about a fixed axis through its center. The dipole is specified by the angular coordinate < ) (the azimuth) so that it constitutes a system of 1 (rotational) degree of freedom. The fractional diffusion equation for the time evolution of the probability density function W(4>, t) in configuration space is given by Eq. (52) which we write here as... 

In order to describe the fractional rotational diffusion, we use the FKKE for the evolution of the probability density function W in configuration angular-velocity space for linear molecules in the same form as for fixed-axis rotators—that is, the form of the FKKE suggested by Barkai and Silbey [30] for one-dimensional translational Brownian motion. For rotators in space, the FKKE becomes... 

The probability density functions (3.14) have the angular dependence. The ip ip value shifts from the vertical z axis in the plane perpendicular to z as the absolute value of mi number increases. 

Figure 3.7 The angular dependence of the s, pz and probability density functions in the...

Figure 10.7 Probability density of die centrifuged oxygen gas as a function of the molecular angle and the free propagation time, that is, die time elapsed since die molecules have been released from die centrifuge. The white dashed line (around 1.5 ps) marks the calculated trajectory of a dumbbell distribution rotating widi die classical rotational frequency of an oxygen molecule with an angular momentum of 39ft. Part of Fig. 4 in Ref. 39.

This is the most stable orbital of a hydrogen-like atom—that is, the orbital with the lowest energy. Since a Is orbital has no angular dependency, the probability density 2 is spherically symmetrical. Furthermore, this is true for all s orbitals. We depict the boundary surface for an electron in an s orbital as a sphere (Figure 1-2). The radial function ensures that the probability for finding the particle goes to zero for r — °°. 

Three components (Q — —1,0,+1) of the multipole moment of rank K = 1 form the cyclic components of the vector. They are proportional to the mean value of the corresponding spherical functions (B.l) for angular momenta distribution in the state of the molecule as described by the probability density p 9,(p). These components of the multipole moments enable us to find the cyclic components of the angular momentum of the molecule ... 

Suppose that the electron is in a 2p state with angular momentum proportional to cos 6 in spherical polar coordinates. The probability density (a ) 2 of such a state would be concentrated near the z-axis, where the length of the radius vector is proportional to cos2 9. Now suppose that the whole system is physically rotated, e.g. by the application of a magnetic field (active rotation) - alternatively the axes may be thought of as rotated in the opposite direction (passive rotation). After rotation the system has a new wave function ip (x) with ip x) 2 concentrated around a displaced axis, but the value of the new wave function at a rotated point must be the same as that of the old wave function at the original point,... 

Clearly, 4>is 2 will not have any angular dependence. That is, it will generate a sphere in three-dimensions. 4>2PJ2, on the other hand, is a periodic function and the probability density is shown in Fig. 10.3. Values of 6... 

Let us now consider how we might represent atomic orbitals in three-dimensional space. We said earlier that a useful description of an electron in an atom is the probability of finding the electron in a given volume of space. The function ij (see Box 1.4) is proportional to the probability density of the electron at a point in space. By considering values of at points around the nucleus, we can define a surface boundary which encloses the volume of space in which the electron will spend, say, 95% of its time. This effectively gives us a physical representation of the atomic orbital, since ij may be described in terms of the radial and angular components R r) and A 0,... 

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