Wave function for the hydrogen atom

Considering the boundary conditions above the solution of the Schrodinger equation for the hydrogen atom formulated in spherical coordinates yields wave functions of the form 

The numbers n, / and m/ known as quantum numbers occur as a result of boundary conditions. The quantum numbers with their allowed values may be summarized as follows  

The function Rni(r) is the radial part of the wave function and its square is related to the probability of finding the electron at any distance r of the nucleus. The functions Qi,mi (0) are the angular part of the wave function which is associated with the quantum numbers I and 

The value of quantum number / is often replaced by a letter  

Thus wave functions are often specified by giving its quantum numbers in the shorthand notation n, I, m,. 

---

To make an informed guess for your first value of ot, you may wish to reread the section on the Bohr theory of the hydrogen atom and the Schroedinger wave functions for the hydrogen atom in a good physical or general chemistry book (see Bibliography). 

FIGURE 13.1 Graphs that have a one-dimensional data space, (a) Radial portion of the wave function for the hydrogen atom in the l.v ground state and 2p excited state. (A) Hypothetical salary chart. 

From electronic structure theory it is known that the repulsion is due to overlap of the electronic wave functions, and furthermore that the electron density falls off approximately exponentially with the distance from the nucleus (the exact wave function for the hydrogen atom is an exponential function). There is therefore some justification for choosing the repulsive part as an exponential function. The general form of the Exponential - R Ey w function, also known as a ""Buckingham " or ""Hill" type potential is... 

As the integers if and / both begin at zero, y = 1,2,3... can of course be identified as the principal quantum number n for the hydrogen atom (see Section 6.6.1). Thus, the quantization of the energy is due to the termination of the series, a condition imposed to obtain an acceptable solution. The associated Laguerre polynomials provide quantitative descriptions of the radial part of the wave functions for the hydrogen atom, as described in Appendix IV. 

The Wave Functions for the Hydrogen Atom. We shall conclude this chapter by discussing the motion of a single electron of mass m and charge — e in the Coulomb field of force witli potential... 

Readers who are still uncomfortable with thinking of functions as vectors should take the time to review this section carefully and do some exercises. These vector spaces are fundamental to our analysis of the hydrogen atom. In particular, we will look at the function space containing the wave functions for the hydrogen atom, and we will work with various subspaces of that function space. 

The wave functions for the hydrogen atom are known exactly. They are functions of the three spatial coordinates of the electron and take their most simple form when we choose these coordinates to be the polar coordinates shown in Figure 8.1 in relation to a set of Cartesian axes. The point at jt y, and z in Cartesian coordinates is fixed by r, the radial distance, OP, from the origin of the coordinate system (always considered positive) 0, the angle between the z axis and the line OP and the angle between the x axis and the projection of OP on the xy plane. 

A Chemical Example the 3s Atomic Radial Wave Function for the Hydrogen Atom... 

Table 6.1. The first few m wave functions for the hydrogen atom...

We call each of these exact one-electron wave functions a molecular orbital (MO), just as we called the exact one-electron wave functions for the hydrogen atom AOs. These exact MOs play a fundamental role in the quantum description of chemical bonding. 

In the case of a system with one degree of freedom no other dynamical quantity (except functions of H only, such as H2) is represented by a diagonal matrix with more degrees of freedom there are other diagonal matrices. For example, the surface-harmonic wave functions for the hydrogen atom... 

The wave functions for the hydrogen atom have the form of a product of a radial function (a function of r only) and the rigid rotor functions. The selection rules for the rotor functions must be the same as those obtained above, namely. 

By an obvious extension of our definition of the wave function for the hydrogen atom, the probability of simultaneously finding electron 1 in dz and electron 2 in dz2 must be given by... 

To write a stationary scattering wave function we argue by a fantihar example. Textbooks write the wave function for the hydrogen atom at a given energy as a... 

Table 1 Expansion of the Ground State Wave Function for the Hydrogen Atom in a Basis Set of Is Gaussian Functions. Total Energy, , is in hartrees the Basis Set Error, AEbs( ) in millihartrees (Relative to —0.5 hartrees). Data taken from Ref. 68...

To represent these solutions (orbitals) on paper is difficult and it is common practice to represent only the angular part. This gives the basic shape of the orbitals and their relative direction in space. The angular parts of some of the wave functions for the hydrogen atom are shown in Fig. 1.4. 

---