The Hydrogen Atom Wavefunctions

Predict the wavelength of light emitted by an excited Li ion (Z = 3) as an electron goes from the n = A state to the = 2 state. Use the mass of the electron in place of the reduced mass (this imparts a very minor 0.008% error in the calculation). 

This wavdength is in the far ultraviolet region of the electromagnetic spectrum. 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

FIGURE 11.17 The energy level diagram for a hydrogen atom, showing the n and quantum numbers for the levels. The quantized energy levels are labeled. Degenerate wavefunctions are 

Orbitals are designated by pairing the value of the principal quantum number and the letter representing the value of ls,2s,2p,3s,3p,3d, and so forth. A numerical subscript can be used to label the trif values of the individual orbitals 2p 1, 2po, 2p+1, and so on. Because the value of n restricts the value of , the first shell has only an s subshell (because can only be 0). The second shell has only s and p subshells (because can only be 0 or 1), and so forth. These restrictions are due to the nature of the mathematical solution of the Schrodinger equation. 

---

P8.5 The hydrogen atom wavefunctions are obtained from the solution of the Schr6dinger equation in Chapter 10. Here we need only the wavefiinction that is provided. It is the square of the wavefimction that is related to the probability (Section 8.4). 

One possibility as a basis set of functions to be used for/is the hydrogen-atom functions. As seen in the case of a helium atom, this is not a particularly good start. The hydrogen-atom wavefunctions do not account for shielding and other affects of the inter-electronic repulsion. A basis set of functions that take this into account is a much better starting point for the calculation. J. C. Slater created such a basis set of functions known as the Slater-type orbitals (STO). The functions have the following general form. 

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 function R(r) is called the radial wavefunction it tells us how the wavefunction varies as we move away from the nucleus in any direction. The function Y(0,c[>) is called the angular wavefunction it tells us how the wavefunction varies as the angles 0 and c > change. For example, the wavefunction corresponding to the ground state of the hydrogen atom ( = 1) is... 

All s-orbitals are independent of the angles 0 and c[>, so we say that they are spherically symmetrical (Fig. 1.31). The probability density of an electron at the point (r,0,ct>) when it is in a ls-orbital is found from the wavefunction for the ground state of the hydrogen atom. When we square the wavefunction (which was given earlier, but can also be constructed as RY from the entries for R and V in Tables 1.2a and 1.2b) we find that... 

The radial parts of the wavefunctions for the hydrogen atom can be constructed from the general form of the associated Laguerre polynomials, as developed in Section 5.5.3. However, in applications in physics and chemistry it is often the probability density that is more important (see Section 5.4.1). This quantity in this case represents the probability of finding the electron in the appropriate three-dimensional volume element. 

As a simple example, consider the hydrogen atom in its ground state, n = 1. The radial part of the wavefunction is given by... 

The understanding of electronic states in atoms is to a great extent based on Schrodinger s solution of the hydrogen-atom problem. These wavefunctions have the general form (Landau and Lifshitz, 1977) ... 

In general, the Slater function is not an exact solution of any Schrodinger equation (except the Is- wavefunction, which is the exact solution for the hydrogen-atom problem). Nevertheless, asymptotically, the orbital exponent C is directly related to the energy eigenvalue of that state. Actually, at large distances from the center of the atom, the potential is zero. Schrodinger s equation for the radial function R(r) is... 

The physical significance of the particle-on-a-sphere wavefunctions is important in connection with the hydrogen atom. Comparing the energy formula (eqn 3.57) with the classical result (eqn 3.44) shows that the angular momentum is... 

---