Wave mechanics

Sections 9.5-9.7: Wave Mechanics (April 28, 1999)

There were many problems with the Bohr model, but it is extremely useful as a teaching device so I continue to use it anyway!

Aside from being unable to predict anything about multi-electron atoms (due to the "3-body" problem), the Bohr model was also inadequate in a number of other ways. It was becoming quite clear that electrons (due to their high velocities) exhibited "wave" characteristics, much like light beams.

DeBroglie postulated that ALL matter exhibited wave characteristics based on the momentum of the particle (momentum = p = mv).
He started from Einstein's theory of relativity:

E = mc²

and also Planck's relationship:

E = hc/l

so that hc/l = mc²,

substituting v for c (the velocity of the particle for the velocity of light) we get:

l = h/mv

The wavelength of an object is inversely proportional to its momentum, p (= mv)... Bottom line: electrons, because they travel SUPERFAST have significant momentum, and hence exhibit short wavelengths.

At the same time Werner Heisenberg, working with Bohr, came up with a mathematical proof that one cannot simultaneously specify BOTH the position and the momentum of an electron.

DpDx > h/4p

The product of the error in momentum and position is always greater than h/4p.

Erwin Schrodinger set about the task of applying wave mechanics to a solution of the energy states of "2-body" systems such as the hydrogen atom. The equation he applied was based on a similar approach which worked for the interaction of light with matter and explained such phenomena as refraction, reflection and diffraction of electromagnetic radiation.

The result should be consistent with the line spectra observed for hydrogen, but also incorporate the newly disovered "wave" properties of an electron.
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The H-hat to the left stands for the Hamiltonian, or total energy operator. It incorporates all potential and kinetic energy terms. The E are the "eigenvalues" or energies. Y are the "wave functions" which are possible solutions given the nature of the Hamiltonian.

The wave functions can be broken up in polar coordinates into two parts, a radial part, and an angular part... Both have qunatum numbers which we will describe shortly. The physical significance of the "wave functions" is simple. The integral of the wavefunction squared over a given volume element:
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is the probability of finding an electron within that volume.

Key point: we no longer talk about an exact "orbit" for the electron, but instead a probability "space." The wavefunction can be considered a mathematical representation of where the electron is, and is often referred to as an "orbital."

The result still contained "quantization" information, but instead of a single quantum number there were three. (A fourth quantum number will be introduced later).

The players are:

n = principle quantum #
    * related to the size and energy of the orbital (described by Y)
        -as n gets larger, so does the "size" of the orbital
          and so does the energy of the orbital.
    * allowed values: any integer greater than or equal to 1.
            n = 1, 2, 3... etc.

l = angular momentum quantum #
* related to the "shape" of the orbital
* allowed values are integers: from 0 to n-1
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Note well: for an "s" orbital the probability of finding an electron near the nucleus is large, whereas for all other types of orbitals it is always 0 at the nucleus. These can be clearly seen from the "shapes" of the orbitals depicted above. When Y² = 0 we call that a node. The p-orbitals have one angular node, d-orbitals have 2 angular nodes, and f-orbitals have 3 angular nodes. .

The value of "l" determines the "shape" of the orbital:

l = 0 ---> "s"
l = 1 ---> "p"
l = 2 ---> "d"
l = 3 ---> "f"
l = 4 ---> "g" etc...

m_l = magnetic quantum #
    * determines "orientation" of the orbital
    * gives the number of orbitals in a subshell
    * can take on integer values of -l... 0... +l
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Each particular combination of n and l are termed "subshells" and are labelled 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p etc... where the number is the principle quantum number while the letter is given by l.

When all orbitals in a subshell (which arise because of all the orientations: values of m_l) are added togethe, they give rise to a spherical distribution. I.e. each "p" orbital may have a figure "8" shape, but when all three are summed, they give rise to the same spherical distribution as described by an "s" orbital.

We come to several conclusions regarding the numbers of orbitals in a particular subshell, and the number of orbitals with a given value of n as illustrated in the table below:
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principle quantum #	possible values for l	subshell label	possible values for m_l
1	0	1s	0
2	0 1	2s 2p	0 -1,0,1
3	0 1 2	3s 3p 3d	0 -1,0,1 -2,-1,0,1,2
4	0 1 2 3	4s 4p 4d 4f	0 -1,0,1 -2,-1,0,1,2 -3,-2,-1,0,1,2,3
5	0 1 2 3 4	5s 5p 5d 5f 5g	0 -1,0,1 -2,-1,0,1,2 -3,-2,-1,0,1,2,3 -4,-3,-2,-1,0,1,2,3,4

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The number of orbitals possible for a given value of n is n². There are 9 orbitals with n = 3, while there are 25 orbitals with n = 5 etc... The number of orbitals in a subshell increases by two as we go from s --> p --> d --> f. There is always only 1 "s" orbital for any principle quantum number, while there are always 3 "p" orbitals for the same value of n. There are always 5 d-orbitals, and 7 f-orbitals.

Click here to move on to the periodic table,
the energies of the subshells, and the
fourth quantum number "spin." 9.8-9.11

Click here to return to section 9.3-9.4