Relative Schrodinger equation values

The first quantum number is the principal quantum number ( ). It describes the energy (related to size) of the orbital and relative distance from the nucleus. The allowed (by the mathematics of the Schrodinger equation) values are positive integers (1, 2, 3, 4, etc.). The smaller the value of n, the closer the orbital is to the nucleus. The number n is sometimes called the atom s shell. 

Equation (10.38) is recognized as the Schrodinger equation (4.13) for the one-dimensional harmonic oscillator. In order for equation (10.38) to have the same eigenfunctions and eigenvalues as equation (4.13), the function Slq) must have the same asymptotic behavior as in (4.13). As the intemuclear distance R approaches infinity, the relative distance variable q also approaches infinity and the functions F(R) and S(q) = RF(R) must approach zero in order for the nuclear wave functions to be well-behaved. As 7 —> 0, which is equivalent to q —Re, the potential U(q becomes infinitely large, so that F(R) and S(q rapidly approach zero. Thus, the function S(q) approaches zero as q -Re and as Roo. The harmonic-oscillator eigenfunctions V W decrease rapidly in value as x increases from x = 0 and approach zero as X —> oo. They have essentially vanished at the value of x corresponding to q = —Re. Consequently, the functions S(iq in equation (10.38) and V ( ) in... 

The quantity is the spin quantum number it may have only the values or —j. The Schrodinger equation in its usual form gives no indication of the existence of the electron spin. However, Dirac has shown that if the Schrodinger equation is cast into a form that satisfies certain requirements of relativity theory, then four quantum numbers, the fourth being the electron spin quantum number, appear in the solution for the hydrogen atom. Thus the spin is a coherent part of the fundamental theory and is not tacked on just to patch things up. 

The value of symmetry arguments in chemical problems arises mainly because the conceptual or computational difficulties involved in applying quantum mechanical methods to complicated molecular systems can often be partially resolved by relatively simple symmetry techniques. However, before these can be applied it is necessary to master the formal methods of handling symmetry transformations. Before discussing the mathematical tools required for this, it is useful to consider one fairly lengthy example of the power of symmetry in simplifying a complex problem. Consider the Schrodinger equation for a system of particles. 

An electron of definite energy cannot be assigned a definite position or a definite path relative to the nucleus it can take any position ftom inside the nucleus to infinity. Instead, for each energy value (set of quantum numbers) the Schrodinger equation yields an expression, called psi squared, that gives the distribution of electrons in an atom or molecule. Applied to the H atom, specifies the probability of finding the electron in a unit of volume at a given distance ft om the nucleus. 

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