Monday, June 28, 2010

More on quantum numbers

In this blog, we examine more closely the quantum numbers introduced briefly in the previous blog. To remind readers the set of quantum numbers was given as: (n, l, m_l) which are identified as: the principal quantum number, the angular momentum quantum number, and the magnetic quantum number, respectively.

There are two physical meanings attendant on n: i) it determines the energy of an orbital (specifically in the H-atom), and (ii) it indicates the average distance of an electron in a particular orbital, to the nucleus, To fix ideas, I show in the accompanying diagram a sketch of one lobe for an electron orbital associated with the (3, 2, +/-2) state in the Hydrogen atom. The key point is the orbital denotes an electron density associated with a probability of finding the electron in some defined space.

In the case shown one must also visualize a symmetrical lobe on the other side (making the whole orbital resemble a dumbell) to make it complete. As one alters the set of quantum numbers (refer to image in previous blog and the numbers on the left side) the electron densities change and so do the probability waves associated with the orbit. To fix ideas I show a second image (Fig. 2) with a series of orbitals for the hydrogen atom, and the sets of quantum numbers that specify each configuration.

Describing orbitals using the set of quantum numbers means knowing the numbering rules applied to each. In the case of the principal quantum number, n, we allow it to have INTEGRAL (non-zero) values: 1, 2, 3, 4 etc.

The physical significance of the angular momentum quantum number (l) is to convey the shape of the probability density cloud or orbital.. The numbering rule for l is directly contingent on the value for n. Thus, for any given n, l must be such that it has integral values from 0 to (n -1). This means if n = 2, then l can have (n- 1) = (2 -1) = 1. But if n =1, then l = (n - 1) = 1 - 1 = 0.

Note that the l-quantum numbers appear more than once for any n >1. Thus, for the n = 2 case, we have TWO values of l occurring: one for l = 0, the other for l = 1. If we go on to n= 3 there are three values of l, for n = 4, four values and so on. One also finds the l-value specified for lettered orbital: s, p, d, f, g, h. The s-orbital is for l=0, the p for l=1, the d for l= 2 and so on. There is no special significance to the letters (apart from the physical meaning we already gave for the quantum number, 1), and they are mainly of historical import- though still retained, for example, in chemistry. (By extension, one also often hears the term "atomic shell" used in chemistry). A collection of orbitals under the same value of n is called a "shell". Thus, for n = 4, we have l=0, l= 1, l=2, l= 3 so comprising the collecton of orbitals:s, p, d and f.

Lastly, there is the magnetic quantum number, usually designated m_l (subscript the same as the angular momentum quantum number) because it is contingent upon it. This quantum number describes the orientation of the orbital in 3-D space. For a given angular momentum quantum number, l, we have integral values of m_l specified as follows:

m_l = -l, (-l +1)....0......(l - 1), +l

Note the above set of m_l numbers is given as a SERIES, e.g. starting with (-l) and terminating at +1. Look at the simplest example for l = 0, then:

m_l = 0.

(Since all terms are zero).

What about l = 1?

Then: m_l = -1, 0, 1

What about l = 2?

We have:

m_l = -2, -1, 0, 1, 2

As a general rubric then, we can use the formula:

N(m_l) = {(2 x l) + 1} to give the total number of m_l numbers.

Netx, we'll look at electron spin and spin quantum number, m_s and where it fits into all this!

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