Application of wave mechanics

Certain features of the Bohr theory are still appropriate to the wave-niechanical picture in particular, the orbits can still be grouped in shells characterized by a principal quantum number n, and the maximum permissible number of electrons in any shell is still 2n2. Not all the electrons in one shell, however, are identical, for those in a shell of principal quantum number n are distributed over n sub-shells characterized by an azimuthal or subsidiary quantum number Z, which can assume any of the values o, 1. (n— 1). Electrons in sub-shells with l = o, 1, 2, 3 are commonly termed s,p, /electrons, respectively, and the state of an electron in respect of its principal and subsidiary quantum numbers is usually symbolized by a figure representing n followed by a letter representing Z. Thus is is an electron in the K shell with / = 0 

Electrons in the same sub-shell have very nearly the same energy, but the energies of those in different sub-shells of the same shell, say the 3p and 3d electrons, are appreciably different. Moreover, the electrons in different sub-shells differ also in that the probability distributions by which they are represented are different in shape, as we shall shortly see. 

The electrons in each sub-shell are distributed in atomic orbitals, the number of which is determined by the magnetic quantum number m. For a give value of /, this quantum number can assume the (2Z+1) possible values — o,. .., 4 /, so that the number of orbitals associated with each sub-shell is as follows  

We shall not normally have occasion to refer to the numerical value of m but it is, nevertheless, essential to keep in mind the number of orbitals in each sub-shell to which it gives rise. 

Finally, each orbital can accommodate only two electrons, and this only if they have oppositely directed spins characterized by the values +1 and — for s, the spin quantum number. Since any orbital is uniquely described by the values of ny l and wz, any electron is uniquely described by the value of n, /, m and s, and no two electrons in an atom can have the same values for all four quantum numbers. This is the formal expression of Pauli s exclusion principle. Again, we shall not normally have occasion to refer to the numerical value of s it will be sufficient to remember that if two electrons occupy the same atomic orbital they must have opposite spins. 

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In Leipzig, Slater pursued the application of wave mechanics to electrons in simple molecules ("quantum chemistry") and in metals ("solid-state physics"). He wrote Percy Bridgman,... 

Continuing our survey of some simple applications of wave mechanics to problems of interest to the nuclear chemist, let us consider the problem of a particle confined to a one-dimensional box (Fig. E.2). This potential is flat across the bottom of the box and then rises at the walls. This can be expressed as ... 

The application of wave mechanics to the nucleus is met by the essential difficulty that the dimensions of the nucleus are of the same order of magnitude (10 cm.) as the diameter of the electron, which... 

In Chapter I we found that curvilinear coordinates, such as spherical polar coordinates, are more suitable than Cartesian coordinates for the solution of many problems of classical mechanics. In the applications of wave mechanics, also, it is very frequently necessary to use different kinds of coordinates. In Sections 13 and 15 we have discussed two different systems, the free particle and the three-dimensional harmonic oscillator, whose wave equations are separable in Cartesian coordinates. Most problems cannot be treated in this manner, however, since it is usually found to be impossible to separate the equation into three parts, each of which is a function of one Cartesian coordinate only. In such cases there may exist other coordinate systems in terms of which the wave equation is separable, so that by first transforming the differential equation into the proper... 

Modem views of atomic structure are, as we have seen, based largely on the applications of wave mechanics to atomic systems. Modern views of molecular structure are based on applying wave mechanics to molecules such studies provide answers as to how and why atoms combine. The Schrodinger equation can be written to describe the behaviour of electrons in molecules, but it can be solved only approximately. Two such methods are the valence bond approach, developed by Heitler and Pauling, and the molecular orbital approach associated with Hund and MuUiken ... 

To assign the value of the constant A, we need to introduce a new idea. In the application of wave mechanics to the description of matter, scientists have learned to associate the square of the wave function with probability. As... 

See also Pauling, L. Wheland, G. W. /. Chem. Phys. 1933,1,362. There is an interesting report that Albert Einstein attended one of Pauling s lectures on the applications of wave mechanics to chemical bonding and said afterward "It was too complicated for me." http //osulibrary. oregonstate.edu/specialcollections/coll/pauling/bond/narrative/page29.html. 

PS. Bagus, B. Liu, A.D. McLean andM. Yoshimine, "Application of wave mechanics to the electronic structure of molecules through configuration interaction" in Wave Mechanics the First Fifty Years. Butterworth. London, 1973. 

The first detailed application of wave mechanics to a stable existent molecule was Heitler and London s 1927 paper in Zeitschrift fur Physik, which served as the cornerstone for many succeeding treatments. The paper employs a perturbation technique to solve the time-independent Schrodinger equation for an electronic wavefunction constructed to represent the H2 molecule. The Hamiltonian for the system is a classical function ... 

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