Quantum numbers and atomic wave functions

The particle in a box example shows how a wave function operates in one dimension. Mathematically, atomic orbitals are discrete solutions of the three-dimensional Schrodinger equations. The same methods used for the one-dimensional box can be expanded to three dimensions for atoms. These orbital equations include three 

Hydrogen Atom Wave Functions Angular Functions Angular factors Real wave functions 

Source Adapted from G. M. Barrow, Physical Chemistry, 5th ed., McGraw-Hill, New York, 1988, p. 450, with permission. 

Note The relations - c )/(2/) = sin 4 and (e + e )l2 = cos 4 can be used to convert the exponential imaginary functions to real trigonometric functions, combining the two orbitals with m = 1 to give two orbitals widt sin 4 and cos j). In a similar fashion, the orbitals with m/ = 2 result in real functions with cos t and sin 4 . These functions have then been converted to Cartesian form by using the functions X = r sin 9 cos j), y = r sin 9 sin ), and z — r cos 9. 

Hydrogen Atom Wave Functions Radial Functions 

The particle-in-a-box example shows how a wave function operates in one dimension. Mathematically, atomic orbitals are discrete solutions of the three-dimensional Schrddinger equations. The same methods used for the one-dimensional box can be expanded to three dimensions for atoms. These orbital equations include three quantum numbers, n, I, and m/. A fourth quantum number, a result of relativistic corrections to the Schrbdinger equation, completes the description by accounting for the magnetic moment of the electron. The quantum numbers are summarized in Table 2.2. Tables 2.3 and 2.4 describe wave functions. 

One feature that should be mentioned is the appearance of i( = N/ ) in the p and d orbital wave equations in Table 2.3. Because it is much more convenient to work with 

Describes orientation of the electron spin (magnetic moment) in space 

Orbitals with different I values are known by the following labels, derived from early terms for different families of spectroscopic lines  

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