Hydrogen, atom, quantum state

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. 

In 1913, Niels Bohr combined elements of quantum theory and classical physics in a treatment of the hydrogen atom. He stated two postulates for an electron in an atom ... 

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... 

Quantum-chemical calculations of PES for carbonic acid dimers [Meier et al. 1982] have shown that at fixed heavy-atom coordinates the barrier is higher than 30kcal/mol, and distance between O atoms is 2.61-2.71 A. Stretching skeleton vibrations reduce this distance in the transition state to 2.45-2.35 A, when the barrier height becomes less than 3 kcal/mol. Meier et al. [1982] have stressed that the transfer is possible only due to the skeleton deformation, which shortens the distances for the hydrogen atom tunneling from 0.6-0.7 A to 0.3 A. The effective tunneling mass exceeds 2mn-... 

In 2-propanoI, the quantum yield for photolytic conversion of benzophenone to the coupled reduction product is 2.0. The reason is that the radical remaining after abstraction of a hydrogen atom from 2-propanol transfers a hydrogen atom to ground-state benzophenone in a nonphotochemical reaction. Because of this process, two molecules of benzophenone are reduced for each one that is photoexcited ... 

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. 

FIGURE 1.28 The permitted energy levels of a hydrogen atom as calculated from Eq. 14. The levels are labeled with the quantum number n, which ranges from 1 (for the lowest state) to infinity (for the separated proton and electron). 

The hierarchy of shells, subshells, and orbitals is summarized in Fig. 1.30 and Table 1.3. Each possible combination of the three quantum numbers specifies an individual orbital. For example, an electron in the ground state of a hydrogen atom has the specification n = 1, / = 0, nij = 0. Because 1=0, the ground-state wavefunction is an example of an s-orbital and is denoted Is. Each... 

The state of an electron in a hydrogen atom is defined by the four quantum numbers n, l, and ms as the value ofn increases, the size of the atom increases. 

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 ... 

The following properties are observed for an unknown element. Identify the element from its properties, (a) The neutral atom has two unpaired electrons, (b) One of the valence electrons in the ground state atom has mt = 4 I. (c) The most common oxidation state is +4. (d) If an electron in a hydrogen atom were excited to the same principal quantum level, n, as the valence electrons in an atom of this element, and fell to the n — 1 quantum level, the photon emitted would have an energy of 4.9 X 10—20 J. 

Hitherto it has been assumed that Tg corresponds to the classical equilibrium (or quantum-mechanical average) distance between the non-bonded atoms in the absence of interaction. It is inherent in the proper application of first-order perturbation theory that the perturbation is assumed to be small. In the case of the hindered biphenyls, however, it is known from the calculations cited in the introduction that the transition state is distorted to a considerable extent. The hydrogen atom does not occupy the same position relative to the bromine atom that it... 

When a hydrogen atom absorbs light, the energy of the photon converts it from the ground state to an excited state. In this process, its electron transfers to an orbital that has a higher energy and a larger principal quantum number. 

The discovery of two other series of emission lines of hydrogen came later. They are named for their discoverers the Lyman series in the ultraviolet range and Paschen series in the infrared region. Although formulas were devised to calculate the spectral lines, the physics behind the math was not understood until Niels Bohr proposed his quantized atom. Suddenly, the emission spectrum of hydrogen made sense. Each line represented the energy released when an excited electron went from a higher quantum state to a lower one. 

Show explicitly for a hydrogen atom in the Is state that the total energy is equal to one-half the expectation value of the potential energy of interaction between the electron and the nucleus. This result is an example of the quantum-mechanical virial theorem. 

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