Probability, and some Quantum Mechanics

Read God’s Debris, by Scott Adams (thanks to Charles L. for the recommendation). It’s a good read. I don’t agree with what it says because I do not believe in the same Theism but it offers up some interesting ideas. What interested me the most was its emphasis on probability.

Specifically, Adams claims that the world is governed by it. Now, this may be what we perceive the world as (a series of events each with a certain probability), but the root cause is not probability. Rather, it’s energetics. For instance, the equilibrium ratio of a state B to a state A of any reversible transformation can be tied back to how energetically favorable said process was. Examples include chemical reactions, diffusion, and heat transport… does G = -RT ln K ring a bell to anyone?

Nevertheless, where concepts of energetics get confusing, probability can work as an abstraction. The ‘classical’ example is quantum mechanics (excuse the pun there, for those who get it :P). By the Heisenburg Uncertainty Principle, one cannot know both the exact position and momentum of a particle at any time. So we use mathematical formulations called wavefunctions, equivalently orbitals, that map out the spatial probability distribution of an electron. One must keep in mind that orbitals are simply mathematical formulations. However, the three-dimensional shape and size, as well as the number of nodes of the orbital (in other words, where it changes “sign”), has immense implications on chemical behavior. The directionality of covalent bonds, ionization potentials, and chemical reactivity are just some examples of how important these mathematical functions are to the physical world.

But wait, you ask: how is the shape and size of an orbital related to its energy? Let’s take a look at where all this come from. It starts with one central equation:
This is, of course, the Schrodinger Equation. H is known as the “Hamiltonian operator” (related to total energy of the system), psi is the wavefunction, and E is an energy state.

Anyone with linear algebra experience should feel a lightbulb light up inside his/her head. The Schrodinger Equation looks very similar to this following, more generic equation:

…The definition of an eigenvalue lambda corresponding to an eigenvector v of a linear matrix transformation A. Relating back to the Schrodinger Equation, we can see that the wavefunction as an “eigenvector” of the Hamiltonian operator corresponding to the eigenvalue E. Thus a wavefunction is simply a mathematical solution that satisfies the Schrodinger equation, where E is an eigenvalue corresponding to an energy level. Only certain eigenvalues exist, so only certain energy levels exist – this is the notion of the quantization of energy that first separated quantum mechanics from classical mechanics.

That aside, we will now delve into the mathematics to further elucidate this connection between energy and probability. The Schrodinger Equation can be rewritten as a partial differential equation, which can be solved by a separation of variables after an appropriate change of variables to spherical coordinates.
(Note the similarities here to solving Fourier’s Heat Equation.) R is the radial term, related to the principle quantum number n, and this determines the “size” of the orbital. P is the azimuthal angular momentum term, related to the angular momentum quantum number l, and this determines the shape of the orbital. F is the magnetic quantum term, related to the magnetic quantum number ml, and this determines the spatial orientation (direction) of the orbital.

Recall that each of these components depends on the original Schrodinger equation – the energetics of it all. And these properties – the simple shape, size, and orientation of orbital probability distributions, determine all of chemistry and chemical bonding. Which explains a good amount of the natural world. Cool stuff! If you consider probability as an abstraction of energy, why yes, Adams is even right on the quantum mechanical level.

Sources
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html
Wikipedia articles: Quantum Number, Hamiltonian, Eigenvalues and Eigenvectors, Wavefunction, Atomic Orbital, Schrodinger Equation
God’s Debris

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