Quantum numbers microstates

The set of microstates of a finite system in quantum statistical mechanics is a finite, discrete denumerable set of quantum states each characterized by an appropriate collection of quantum numbers. In classical statistical mechanics, the set of microstates fonn a continuous (and therefore infinite) set of points in f space (also called phase space). 

Table 3.4 Quantum Numbers n i and ms for Two Electrons in Configuration p Term Symbol Assignment9 l, Microstate Number, and...

A multielectron atom can exist in several electronic states, called microstates, which are characterized by the way the electrons are distributed among the atomic orbitals. The number of microstates for a free atom with a valence shell consisting of a set of degenerate orbitals with orbital angular momentum quantum number I housing n electrons is given by ... 

The 2p electrons are not independent of each other, however the orbital angular momenta (characterized by mi values) and the spin angular momenta (characterized by m, values) of the 2p electrons interact in a manner called Russell-Saunders coupling or LS coupling. The interactions produce atomic states called microstates that can be described by new quantum numbers ... 

Each set of possible quantum numbers (such as ) is called a microstate. 

The next step is to tabulate the possible microstates. In doing this, we need to take two precautions (1) to be sure that no two electrons in the same microstate have identical quantum numbers (the Pauli exclusion principle applies) and (2) to count only the unique microstates. For example, the microstates and O " , and 0 0" in a configuration are duplicates and only one of each pair will be listed. 

We have now seen how electronic quantum numbers nti and may be combined into atomic quantum numbers A/, and Ms, which describe atomic microstates. M and Ms, in turn, give atomic quantum numbers L, S, and J. These quantum numbers collectively describe the energy and symmetry of an atom or ion and determine the possible transitions between states of different energies. These transitions account for the colors observed for many coordination complexes, as will be discussed later in this chapter. 

Quantum numbers L and S describe collections of microstates, whereas Mi and Ms describe the microstates themselves. L and S are the largest possible values of Mi and Ms. Mi is related to L much as m is related to /, and the values of Ms and are similarly related ... 

To ensure that all microstaics have been written, the total number N. of microstates associated with an electronic configuration, A, having. r electrons in an orbital set with an azimuthal quantum number, I, is... 

Let us now consider the two configurations 2/>3p and 2p2. In the first case, our freedom to assign quantum numbers mt and ms to the two electrons is unrestricted by the exclusion principle since the electrons already differ in their principal quantum numbers. Thus microstates such as (1+, 1+) and (0, 0 ) are permitted. They are not permitted for the 2p2 configuration, however. Secondly, since the two electrons of the 2p3p configuration can be... 

Each set of possible quantum numbers, such as 1+0, which communicates each unique possible coupling of magnetic fields of the electrons, is called a microstate. 

Just as the quantum number ttii describes the z-component of the magnetic field due to an electron s orbital motion, the quantum number Mi describes the z-component of the magnetic field associated with a microstate. Similarly, describes the magnetic field due to an electron s spin in a reference direction (usually defined as the z direction), and Ms describes the analogous component of the magnetic field produced by electron spin for a microstate. 

With sets of quantum numbers in hand, the electronic states (microstates) that are possible for a given electronic configuration can be determined. This is best achieved by constructing a table of microstates, remembering that ... 

An electron in an s atomic orbital must have 1=0 and /w/ = 0, and for each electron, can be +jor-j. The n configuration is described in Table 21.4. Applying the Pauli exclusion principle means that the two electrons in a given microstate must have different values of m i.e. f and J. in one row in Table 21.4. A second arrangement of electrons is given in Table 21.4, but now we must check whether this is the same as or different from the first arrangement. We cannot physically distinguish the electrons, so must use sets of quantum numbers to decide if the microstates (i.e. rows in the table) are the same or different ... 

The third law is a consequence of the statistical nature of entropy as reflected in Boltzmann s formula (Equation 8.1), which relates the entropy to W, the number of molecular quantum states (microstates) consistent with the macroscopic conditions. At r = 0, there is no available thamal energy, and the thermodynamically most stable state is the lowest possible energy state (the ground state). In general, this state is unique, so W = 1 for a system at 0 K. From Boltzmann s formula we then have... 

Solution. The (3<2) C4 1) problem is slightly different from the Qtd y problem. Both electrons are d electrons with 1—1, but one has = 3 and one has = 4. Thus, for example, the (2, 2) microstate does not violate the Pauli principle, since the n quantum numbers differ. The bookkeeping is simplified by adding a subscript 4 to the mi value for the 4d electron. 

Next we look for violations of the Pauli exclusion principle. This leads us to strike out rows 1,4,17,20,33, and 36, labeling them P for Pauli. Our remaining microstates number 15 and are reassembled in Table 5-4, along with values of the quantum numbers for z components of the relevant angular momentum vectors for individual electrons as well as for their sum. 

Macroscopic states involve variables that pertain to the entire system, such as the pressure P, the temperature T, and the volume V. For a fluid system of one substance and one phase, the equilibrium macrostate is specified by only three variables, such as P, T, and V. If we assume that classical mechanics is an adequate approximation, the microstate of such a system is specified by the position and velocity of every particle in the system. If quantum mechanics must be used for a dilute gas, there are several quantum numbers required to specify the state of each molecule in the system. This is a very large number of independent variables or a very large number of quantum numbers. Statistical mechanics is the theory that relates the small amount of information in the macrostates and the large amount of information in the microstates. 

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