Hydrogen atom probability distribution

Humberston and Wallace, 1972), is shown in Figure 6.4. Also shown there is the distribution function obtained using the Born approximation, in which neither the positron nor the atomic wave function is modified by the interaction. This latter curve therefore represents the momentum distribution of the electron in the undistorted hydrogen atom. The distribution function for the accurate wave function is narrower than that for the undistorted case because the positron attracts the electron towards itself and away from the nucleus, thereby enhancing the probability of low values of the momentum of the pair. 

The reason a single equation = ( can describe all real or hypothetical mechanical systems is that the Hamiltonian operator H takes a different form for each new system. There is a limitation that accompanies the generality of the Hamiltonian and the Schroedinger equation We cannot find the exact location of any election, even in simple systems like the hydrogen atom. We must be satisfied with a probability distribution for the electron s whereabouts, governed by a function (1/ called the wave function. 

Another, more common way of showing the electron distribution in the ground state of the hydrogen atom is to draw the orbital (Figure 6.4b) within which there is a 90% chance of finding the electron. Notice that the orbital is spherical, which means that the probability is independent of direction the electron is equally likely to be found north, south, east, or west of the nucleus. 

FIGURE 1.42 The radial distribution functions for s-, p-, and cf-orbitals in the first three shells of a hydrogen atom. Note that the probability maxima for orbitals of the same shell are close to each other however, note that an electron in an ns-orbital has a higher probability of being found close to the nucleus than does an electron in an np-orbital or an nd-orbital. 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

It will be assumed for the moment that the non-bonded atoms will pass each other at the distance Tg (equal to that found in a Westheimer-Mayer calculation) if the carbon-hydrogen oscillator happens to be in its average position and otherwise at the distance r = Vg + where is a mass-sensitive displacement governed by the probability distribution function (1). The potential-energy threshold felt is assumed to have the value E 0) when = 0 and otherwise to be a function E(Xja) which depends on the variation of the non-bonded potential V with... 

The formation of a more-stable free radical increases the selectivity of the reaction. For this reason, the replacement of a particular hydrogen atom by a halogen isn t simply a matter of probability. In propane, replacement of one of the hydrogen atoms on the central carbon should occur one-fourth (Va) of the time. (You may want to draw this reaction to see why this is true.) However, chlorination shows a distribution where replacement occurs at the second carbon about three-fourths of the time, and for bromination, the replacement is almost exclusively on the central carbon atom. Table 2-1 indicates the relative selectivity of chlorine and bromine. 

The hydrogen atom orbitals are functions of three variables the coordinates of the electron. Their physical interpretation is that the square of the amplitude of the wave function at any point is proportional to the probability of finding a particle at that point. Mathematically, the electron density distribution is equal to the square of the absolute value of the wave function ... 

Fig. 2. The potential energy curve for the motion of hydrogen atoms in oxalic acid dihydrate. The zero point energy levels (exaggerated) and the probability distribution curves for H (full line) and D (broken line) are shown...

Fig. 1-2.—The wave function u, its square, and the radial probability distribution function 47rrVi fo the normal hydrogen atom.

Reaction at a carbon atom may be inhibited by the proximity of neighboring groups. Steric effects of hydrogen atoms on carbon atoms, five atoms removed from the site of attack, are important. This principle, called by Newman (150) the rule of six, has been used (63) to explain yield distribution in hydrocarbon oxidation. Steric effects are probably important in the high resistance of some tertiary C—H bonds to attack (63,... 

In ihe quantum-mechanical description of a hydrogen atom. Ihe radial portion of the probability densily distribution is the same in all directions from Ihc nucleus, hul only for the case I = 0 is Ihe magnitude of the dislrihulion the same in all radial directions. For all other values of /. the magnitude of the distribution is a function of the angular direction, defined... 

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