Atoms stationary states

Ket notation is sometimes used for functions in quantum mechanics. In this notation, the function / is denoted by the symbol j/) /—1/>. Ket notation is convenient for denoting eigenfunctions by listing their eigenvalues. Thus nlm) denotes the hydrogen-atom stationary-state wave function with quantum numbers , /, and m. 

It was emphasized in Chapter 6 that the definition of an atomic stationary state property is determined by the form of the atomic stationary state functional fl]. In precisely the same manner, the definition of an atomic property in the general time-dependent case is determined by the form of the atomic Lagrangian integral 2,t]. In both the stationary-state and... 

Calculate the uncertainty AL for these hydrogen-atom stationary states (a) 2p/,... 

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that contain more than one particle. Apart from the hydrogen atom, the stationary-state energies caimot be calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and widely used approximation in chemistry is the independent-particle approximation, which can take several fomis. Conuiion to all of these is the assumption that the Hamiltonian operator for a system consisting of n particles is approximated by tlie sum... 

All the previous discussion in this chapter has been concerned with absorption or emission of a single photon. However, it is possible for an atom or molecule to absorb two or more photons simultaneously from a light beam to produce an excited state whose energy is the sum of the energies of the photons absorbed. This can happen even when there is no intemrediate stationary state of the system at the energy of one of the photons. The possibility was first demonstrated theoretically by Maria Goppert-Mayer in 1931 [29], but experimental observations had to await the development of the laser. Multiphoton spectroscopy is now a iisefiil technique [30, 31]. 

In Figure 2.2(a) states m and n of an atom or molecule are stationary states, so-called because they are time-independent. This pair of states may be, for example, electronic, vibrational or rotational. We consider the three processes that may occur when such a... 

The modern theory of the behavior Of matter, called quantum mechanics, was developed by several workers in the years 1925-1927. For our purposes the most important result of the quantum mechanical theory is that the motion of an electron is described by the quantum numbers and orbitals. Quantum numbers are integers that identify the stationary states of an atom the word orbital means a spatial description of the motion of an electron corresponding to a particular stationary state. 

All atoms have stationary states and can hold only particular values of energy. 

According to this scheme an atom of sulphur, for example, with sixteen electrons, would have an electronic configuration of 2, 4, 4, 6 (Bohr [1923]). The main feature of the building-up procedure was Bohr s assumption that the stationary states would also exist in the next atom, obtained by the addition of a further electron. He also assumed that the number of stationary states would remain unchanged apart from any additional states of the newly introduced electron. In other words the assumption was one of the existence of sharp stationary states, and their retention on adding both an electron and a proton to an atom. To quote Bohr ... 

It should be added that the stationary states in the newly obtained atom are not identical, since the accompanying addition of a proton causes a contraction in the size of the electron orbits. 

For the purposes of fixing the stationary states we have up to this point only considered simply or multiply periodic systems. However the general solution of the equations frequently yield motions of a more complicated character. In such a case the considerations previously discussed are not consistent with the existence and stability of stationary states whose energy is fixed with the same exactness as in multiply periodic systems. But now in order to give an account of the properties of the elements, we are forced to assume that the atoms, in the absence of external forces at any rate always possess sharp stationary states, although the general solution of the equations of motion for the atoms with several electrons exhibits no simple periodic properties of the type mentioned (Bohr [1923]). 

Pauli s original version of the exclusion principle was found lacking precisely because it ascribes stationary states to individual electrons. According to the new quantum mechanics, only the atomic system as a whole possesses stationary states. The original version of the exclusion principle was replaced by the statement that the wavefunction for a system of fermions must be antisymmetrical with respect to the interchange of any two particles (Heisenberg [1925], Dirac [1928]). 

Soon after Bohr developed his initial configuration Arnold Sommerfeld in Munich realized the need to characterize the stationary states of the electron in the hydrogen atom by. means of a second quantum number—the so-called angular-momentum quantum number, Bohr immediately applied this discovery to many-electron atoms and in 1922 produced a set of more detailed electronic configurations. In turn, Sommerfeld went on to discover the third or inner, quantum number, thus enabling the British physicist Edmund Stoner to come up with an even more refined set of electronic configurations in 1924. 

With time-dependent computer simulation and visualization we can give the novices to QM a direct mind s eye view of many elementary processes. The simulations can include interactive modes where the students can apply forces and radiation to control and manipulate atoms and molecules. They can be posed challenges like trapping atoms in laser beams. These simulations are the inside story of real experiments that have been done, but without the complexity of macroscopic devices. The simulations should preferably be based on rigorous solutions of the time dependent Schrddinger equation, but they could also use proven approximate methods to broaden the range of phenomena to be made accessible to the students. Stationary states and the dynamical transitions between them can be presented as special cases of the full dynamics. All these experiences will create a sense of familiarity with the QM realm. The experiences will nurture accurate intuition that can then be made systematic by the formal axioms and concepts of QM. 

One can hence think of (normal-mode composition factor) ej = ejaSja as the fractional involvement of atom j in normal mode a.The dimensionless vector eja also specifies the direction of the motion of atom j in the ot-th normal mode. Interestingly, the mode composition factors are also related to the magnitude of the atomic fluctuations. In a stationary state ) of a harmonic system, the mean square deviation (msd) of atom j from its equilibrium position may be expressed as a sum over modes of nonzero frequency ... 

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent Schrodinger equation ... 

As was already noted in [9], the primary effect of the YM field is to induce transitions (Cm —> Q) between the nuclear states (and, perhaps, to cause finite lifetimes). As already remarked, it is not easy to calculate the probabilities of transitions due to the derivative coupling between the zero-order nuclear states (if for no other reason, then because these are not all mutually orthogonal). Efforts made in this direction are successful only under special circumstances, for example, the perturbed stationary state method [64,65] for slow atomic collisions. This difficulty is avoided when one follows Yang and Mills to derive a mediating tensorial force that provide an alternative form of the interaction between the zero-order states and, also, if one introduces the ADT matrix to eliminate the derivative couplings. 

Solution of (12) gives the complete non-relativistic quantum-mechanical description of the hydrogen atom in its stationary states. The wave function is interpreted in terms of... 

From Bohr s postulates, it is possible to derive the energies of the possible stationary states responsible for the radiation that is absorbed or emitted by an atom consisting of a single electron and nucleus. The specification of these states permits one to then compute the frequency of the associated electromagnetic radiation. To begin, one assumes the charge on the nucleus to be Z times the fundamental electronic charge, e, and that Coulomb s law provides the attractive force, F, between the nucleus and electron ... 

A dispute123,125> 126 over the magnitude of k2h (Table 8) seems to have been settled in favour of Fenimore s value by recent observations107 109, 127. The finding of Gutman et al,107 of a stationary state in O atoms for the decomposition of 4 % N20 in Ar at temperatures between 1800° and 2500 °K certainly supports Fenimore s rate coefficient. Olschewski et al.109 have clearly demonstrated in their shock-tube study that departure from a steady state in O atoms only occurs at very low N20 concentrations, viz for T > 2000 °K at 0.02 % in Ar. This observation suggests, in fact, that Fenimore s value of k2b represents a lower limit for the... 

The atom has only specific, allowable energy levels, called stationary states. Each stationary state corresponds to the atom s electrons occupying fixed, circular orbits around the nucleus. 

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