Hydrogen-like atom quantum numbers

The energy levels of the hydrogen-like atom depend only on the principal quantum number n and are given by equation (6.48), with replaeed by ao, as... 

The wave functions nlm) for the hydrogen-like atom are often called atomic orbitals. It is customary to indicate the values 0, 1, 2, 3, 4, 5, 6, 7,. .. of the azimuthal quantum number / by the letters s, p, d, f, g, h, i, k,. .., respectively. Thus, the ground-state wave function 100) is called the Is atomic orbital, 200) is called the 2s orbital, 210), 211), and 21 —1) are called 2p orbitals, and so forth. The first four letters, standing for sharp, principal, diffuse, and... 

The hydrogen-like atomic energy levels are given in equation (6.48). If n and 2 are the principal quantum numbers of the energy levels E and E2, respectively, then the wave number of the spectral line is... 

Show that in a hydrogen-like atom of nuclear charge Ze the average distance of the electron from the nucleus, in the stale described by quantum numbers /, >i, is... 

When the wave equation for a hydrogen-like atom is solved in the most direct way for orbitals with the angular momentum quantum number / = 3, the following results are obtained for the purely angular parts (i.e., omitting all numerical factors) ... 

Although wave equations are readily composed for more-electron atoms, they are impossible to solve in closed form. Approximate solutions for many-electron atoms are all based on the assumption that the same set of hydrogen-atom quantum numbers regulates their electronic configurations, subject to the effects of interelectronic repulsions. The wave functions are likewise assumed to be hydrogen-like, but modified by the increased nuclear charge. The method of solution is known as the self-consistent-field procedure. 

The second class of hydrogen-like atoms is called Rydberg atoms. A Rydberg atom is an ordinary atom in which one electron has been elevated to a very high quantum state. The energy states of atoms are identified with the quantum number n, called the principal quantum number. The ground state, or lowest state, is the n = state, which is where atoms spend most of their time. The first excited state is the n = 2 energy state, the second excited state is = 3, and so on. 

For the higher quantum numbers the relationship between the energy values of the orbitals is more complicated (see Figure 7). Thus for example, the 4J orbital ( == 4, / = m) is more stable than the orbital (n = 3, / = 2). This complexity docs not permit a construction of the electronic distribution of the elements on the basis only of the analogy to a hydrogen like atom and it is necessary for each element to use the spectroscopic data to determine the electronic states. The sequence of the distribution thus obtained can, with only a few exceptions, be expressed by the data given in Figure 7. 

In general, for a particular value of the principal quantum number, there cannot be more than two s, six />, ten d and fourteen/ electrons and the total possible number of electrons for a given value of n is therefore equal to 2n. Thus the introduction of the concept of spin leads to the doubling of the number of possible electronic states (equation 1.39). The possible distribution of electrons for a hydrogen like atom is shown in Table V and it is seen that the series 2, 8, 18, 32,. . . which has arisen from the application of the Pauli principle is in agreement with the numbers of elements occurring in the periodic table of Mendeleeff. The electron shells with values for the principal quantum number i, 2, 3, 4, etc are often referred to as the /if, L, Af, JV, etc shells. 

The expectation values of various powers of the radial variable r for a hydrogen-like atom with quantum numbers n and I are given by equation (6.69)... 

The standard approach to solving the Schrodinger equation for hydrogen-like atoms involves transforming it from Cartesian (x, y, z) to polar coordinates (r, 6, (p), since these accord more naturally with the spherical symmetry of the system. This makes it possible to separate the equation into three simpler equations, /(r) = 0, f(9) = 0, and /(0) = 0. Solution of the f[r) equation gives rise to the n quantum number, solution of the f(9) equation to the / quantum number, and solution of the f(quantum number. For each specific n = n, l = V and... 

Sommerfeld s formula for the energy of hydrogen-like atoms is derived under the assumptions of relativistic motion under an inverse square attractive force, subject to the above quantum postulates. It exhibits only two quantum numbers, n and kt The third, m, appears explicitly if the atom is situated in a... 

These individual functions R, , and give rise to the three orbital quantum numbers n, i, and m. We have seen that solutions of the Schrodinger equation are possible only for certain values of the total energy E. For hydrogen-like atoms, the permitted total energy values are given by the equation... 

For hydrogen-like atoms, which have one electron, the energy depends only on the principal quantum number n the energies of the 2s and 2p orbitals are the same. For atoms having more than one electron, however, the energy also depends on the value of I... 

When procedures such as this are adopted, it is still possible to assign to each orbital four quantum numbers n, I, m, and 5, the latter being for spin. However, whereas for hydrogen-like atoms the energy of the electron depends only on the principal quantum number n, for atoms containing more than one electron the energy depends also on I This is illustrated in Figure 1.8, which shows schematically the relative... 

Rydberg constant (R - The fundamental constant which appears in the equation for the energy levels of hydrogen-like atoms i.e., E = hcR. 2 JrP-, where h is Planck s constant, c the speed of light, Z the atomic number, (Xthe reduced mass of nucleus and electron, and n the principal quantum number (n = 1,2,. ..). 

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