Quantum numbers types

Worked example 1.3 Quantum numbers types of orbital... 

Wliat does one actually observe in the experunental spectrum, when the levels are characterized by the set of quantum numbers n. Mj ) for the nonnal modes The most obvious spectral observation is simply the set of energies of the levels another important observable quantity is the intensities. The latter depend very sensitively on the type of probe of the molecule used to obtain the spectmm for example, the intensities in absorption spectroscopy are in general far different from those in Raman spectroscopy. From now on we will focus on the energy levels of the spectmm, although the intensities most certainly carry much additional infonnation about the molecule, and are extremely interesting from the point of view of theoretical dynamics. 

We have described here one particular type of molecular synnnetry, rotational symmetry. On one hand, this example is complicated because the appropriate symmetry group, K (spatial), has infinitely many elements. On the other hand, it is simple because each irreducible representation of K (spatial) corresponds to a particular value of the quantum number F which is associated with a physically observable quantity, the angular momentum. Below we describe other types of molecular synnnetry, some of which give rise to finite synnnetry groups. 

Cartesian Gaussian-type orbitals (GTOs) Jfa.i.f( ( characterized by the quantum numbers a, b and c, which detail the angular shape and direction of the orbital, and the exponent a which governs the radial size . 

As is to be expected, inherent disorder has an effect on electronic and optical properties of amorphous semiconductors providing for distinct differences between them and the crystalline semiconductors. The inherent disorder provides for localized as well as nonlocalized states within the same band such that a critical energy, can be defined by distinguishing the two types of states (4). At E = E, the mean free path of the electron is on the order of the interatomic distance and the wave function fluctuates randomly such that the quantum number, k, is no longer vaHd. For E < E the wave functions are localized and for E > E they are nonlocalized. For E > E the motion of the electron is diffusive and the extended state mobiHty is approximately 10 cm /sV. For U <, conduction takes place by hopping from one localized site to the next. Hence, at U =, )J. goes through a... 

Make a table listing the principal quantum numbers (through three), the types of orbitals, and the number of orbitals of each type. 

It is not easy to see why the authors believe that the success of orbital calculations should lead one to think that the most profound characterization of the properties of atoms implies such an importance to quantum numbers as they are claiming. As is well known in quantum chemistry, successful mathematical modeling may be achieved via any number of types of basis functions such as plane waves. Similarly, it would be a mistake to infer that the terms characterizing such plane wave expansions are of crucial importance in characterizing the behavior of atoms. 

Self-Test 1.10A The three quantum numbers for an electron in a hydrogen atom in a certain state are n = 4, l = 2, and ml = —1. In what type of orbital is the electron located ... 

Transverse magnetization represents a particular type of coherence involving a change in quantum number p of 1. Each coherence a is equal to the difference in magnetic quantum numbers of the nuclei r and 5, i.e., the coherence order is M,—M and pulses cause transitions to occur... 

Single-quantum coherence Coherence between states whose total quantum numbers differ by 1. The only type of coherence that can be observed directly. 

We consider a nuclear wave function describing collisions of type A + BC(n) AC(n ) + B, where n = vj, k are the vibrational v and rotational j quantum numbers of the reagents (with k the projection of j on the reagent velocity vector of the reagents), and n = v, f, k are similarly defined for the products. The wave function is expanded in the terms of the total angular momentum eigenfunctions t X) [63], and takes the form [57-61]... 

To appreciate this point somewhat better it is useful to compare three types of Gaussian basis sets, (a) a set of Gaussians with common orbital exponents (for one 1) but a sequence of principle quantum-numbers... 

A particle possesses an intrinsic angular momentum S and an associated magnetic moment Mg. This spin angular momentum is represented by a hermitian operator S which obeys the relation S X S = i S. Each type of partiele has a fixed spin quantum number or spin s from the set of values 5 = 0, i, 1,, 2,. .. The spin s for the electron, the proton, or the neutron has a value The spin magnetie moment for the electron is given by Mg = —eS/ nie. 

Depending on the permitted values of the magnetic quantum number m, each subshell is further broken down into units called orbitals. The number of orbitals per subshell depends on the type of subshell but not on the value of n. Each orbital can hold a maximum of two electrons hence, the maximum number of electrons that can occupy a given subshell is determined by the number of orbitals available. These relationships are presented in Table 17-5. The maximum number of electrons in any given energy level is thus determined by the subshells it contains. The first shell can contain 2 electrons the second, 8 electrons the third, 18 electrons the fourth, 32 electrons and so on. 

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