Indeed, Schrödinger's was the 1st "Picture" of quantum mechanics. It rested on the nature of all wave equations. In the absence of forces, waves are simple sinusoidal motion whose spatial extent exemplifies its wavelength but whose travel by a reference point gives evidence of its frequency.

And deBroglie had given the apparent connection between matter's momentum and it's (e.g., electron diffraction) wavelength.
While Planck had made the connection between frequency and energy. All that was left was to believe that it all came together in (simple) matter waves whose components at least had a seductively simple form and consequence:

Where the barred h is just Planck's Constant divided by two pi.

The consequence stemmed from the additivity of waves. It's that (via the Davisson-Germer expt.) which triggered deBroglie, and it's that which so dismayed Einstein. For the wave is a pure sine of infinite extent given a particular momentum p and energy E. To localize the wave, one must add sines of other p (and E) such that their sum is mutually destructive at large distances but remains constructive where the particle is to be localized.

This is easily done (see "wave packets" in QM texts), but carries with it wave mechanic's penalty...there is no longer a single p (or E) but rather a distribution of them in the superposition. As one tries to confine a wave spatially, it's momentum becomes uncertain without limit. Leaving well enough alone (the original sine) gives completely certain p but completely uncertain position (the wave has infinite extent).

Heisenberg (may have [snicker]) quantified that in these famous Uncertainty Relations. The first tells us we cannot know position and momentum simultaneously. (This yields zero point energies forbidding an oscillator to ever be at rest...even at Absolute Zero.) The second forbids detailed knowledge of the energy states of newly-created systems...chemical products, for example. Only at times distant from their creation will they have "stationary state" energies. (This gives us "collisional broadening" in spectra. New states arise with almost every collision in a liquid, for example. It's spectrum is not one of lines but of mush.)

But Schrödinger's waves hold another critical importance for the rest of this course.

Dealing with only the time-independent part of the wave for our purposes at the moment, the differential equation governing wave motion is (remember the sine above):

The potential function, V(x), establishes boundaries for classical motion. It can never be the case that E<V because the kinetic energy, K, would have to be negative there. So positions, X, where V(X)=E, are "turning points" where classical trajectories "bounce" off the potential barrier. While the 3rd Picture of quantum mechanics (see below) ensures that wave functions are more brave than classical paths in this regard (tunnelling), even they are rapidly falling off to nothing in the vicinity of V(X)=E. But only those with the correct phase can do so, and they must have such a correct phase at all possible turning points.

Only waves of unique total energies will be so clever. So it is the constraining potentials in quantum mechanics which mandate discrete energy levels! Where potentials are absent, as in a freely translating particle, even quantum mechanical energies can be continuous. But given any bounding potential, discrete stationary states arise automatically.

In PChem, we derived or rationalized many of these for molecular modes of motion. Here is such as set. Some notion of the order of magnitudes of these is important. Under ordinary conditions, at thermal energies there are many more than Avogadro's Number of translational energy levels available to any molecule. There are hundreds of rotational levels so available. There will likely be only one or two available levels to any vibration. And there will definitely be only one electronic level, the ground state of course, available at ordinary temperatures.

The rotational energy levels are for symmetrical top molecules with moments of inertia along the symmetry axis (subscript s) and perpendicular to it (subscript p). For diatomic or fully spherical molecules, K=0 and the expected formula results. General (non-symmetric) molecules have no algebraic expression like this. Likewise there are no simple algebraic expressions for electronic energies beyond the hydrogen atom.

We will need these energy levels later in the course to calculate thermodynamic properties in many cases more accurately than they can be measured!

And we'll need one thing more...degeneracies. We must know how many wavefunctions have the same energy because our "state counts" will be of valid wavefunctions not different energy levels. We recall diatomic rotation has a 2J+1 degeneracy. Triplet S atoms are triply degenerate. Molecules with symmetry have multiple vibration modes with the same frequencies; hence their levels are degenerate as well. And we'll find that translational degeneracies will rise as the square of molecular velocities. All of this and more (e.g., rotational symmetry factors) will fall out of quantum into statistical mechanics and have thermodynamic consequences.

---

There are other "Pictures" of quantum mechanics. In PChem you've already dealt with one: Heisenberg's Matrix Mechanics. It recognizes that the quantum Hamiltonian (the lefthand side of the Schrödinger Equation above) applied to the correct wavefunction simply returns it scaled by its energy. So, as an operator, the Hamiltonian can "pick out" valid wavefunctions from their superposition in some sense. You've used it in Hückel Theory and for extinction coefficients in spectra.

The 3rd "picture" is Feynman's path integrals. Instead of hewing to the path foretold by the minimum in the Classical Action integral, quantum "trajectories" are centered on that path but explore alternatives with lesser and lesser likelihood the further afield from the classical trajectory they go. Because the quantum possibilities are thus "fuzzy" about the path, it's possible to find a quantal "particle" across a turning point forbidden to its classical brother. Once there, it can meander near the classically allowed paths (if any) on the other side...rather like Alice (tunnelling) Through the Looking Glass.

There is even a 4th "picture" called Creation/Annihilation Operators, but it doesn't illuminate much for chemists. So I'll not burden you with it.

Chris Parr
University of Texas at Dallas
Programs in Chemistry, Room BE3.506
P.O. Box 830688  M/S BE2.6 (for snailmail)
Richardson, TX  75083-0688

   Voice: (214) 883-2485
     Fax: (214) 883-2925
     BBS: (214) 883-2168 (HST) or -2932 (V.32bis)
Internet: parr@utdallas.edu  (Click on that address to send Dr. Parr e-mail.)