Hydrogen-like atom expectation values

The expectation values of powers and inverse powers of r for any arbitrary state of the hydrogen-like atom are defined by... 

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)... 

Figure 1.14 Expectation value of the electric field strength for the lowest-lying states of a hydrogen-like atom in the range Z = 1-92. Electron wave functions for extended nuclear-charge distributions are employed.

For non-pointlike dipoles, one may expect this limiting value to be less important, since the essence of the problran is binthng an electron by a positive charge. This, however, happrais even for marginally small positive charges (see the hydrogen-like atom). 

These familiar relativistic one-electron operators, whose expectation values often served to define measures for relativistic effects, are only an approximation to the Dirac Hamiltonian of hydrogen-like atoms discussed in chapter 6. Nevertheless, they are also employed in the context of many-electron systems where the one-electron part in / of Eq. (11.1), i.e., Vnuo is approximated solely by the 2 x 2-analogous expressions for Ejoj - -... 

Calculate r), the expectation value of r for the 2s and 2p states of a hydrogen-like atom. Comment on your answer. 

Calculate the expectation values (1 /r> and V) for a hydrogen-like atom in the Is state. 

Calculate the expectation value of the square of the speed of the electron in a hydrogen-like atom with Z = 26 (an Fe + ion) in the li state, and from this calculate the root-mean-square speed. Compare this speed with the speed of light and with the root-mean-square speed of the electron in a hydrogen atom from Example 17.9. 

You have very likely seen stationary states before, although the name might be new to you. The orbitals in a hydrogen atom (Is, 2pz, and so forth) are all stationary states, as we will discuss in Section 6.3. The probability distribution P(x) and the expectation values of all observables are constant in time for stationary states. 

Similar quantitative estimation of the role of the relativistic effects can be done on the basis of the quantum mechanics by the substitution of the velocity v by its expectation value in the given state. With the virial theorem one obtains for the hydrogen-like ions with the atomic number Z ... 

Because particles have wavelike properties, we cannot expect them to behave like pointlike objects moving along precise trajectories. Schrodinger s approach was to replace the precise trajectory of a particle by a wavefunction, i]i (the Greek letter psi), a mathematical function with values that vary with position. Some wavefunctions are very simple shortly we shall meet one that is simply sin x when we get to the hydrogen atom, we shall meet one that is like e x. 

The three carbon atoms of the cyclopropane ring lie in a plane. Therefore the angle strain is expected to be considerable because each C-C-C valence angle must be deformed 49.5° from the tetrahedral value. It is likely that some relief from the strain associated with the eclipsing of the hydrogens of cyclopropane is achieved by distortion of the H-C-H and H-C-C bond angles ... 

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