So far, this could be just a bunch of abstract math. These rules apply to any differential equation of the kind we are discussing. But we are interested in what is physically represented by the quantities in the equation. Well, as we’ve already alluded, Schroedinger was able to show that the right side of his equation corresponds to the total energy of the particle multiplied by the wave function for that particle.
Recall now that the total energy is just the sum of potential energy and energy associated with the motion of the particle (the kinetic energy). The potential energy shows up explicitly in the form of the potential energy function U. This implies that the remaining term on the left side represents the particle’s kinetic energy.
In a very real sense, masked by all the funny math symbols, Schroedinger’s equation is just a statement of this basic fact: total energy equals kinetic energy plus potential energy. Energy is the dominant theme in all of this, and even though a particle’s position and velocity may be fuzzy, it is still valid to think of particles having definite amounts of total energy.
Schroedinger’s equation has another interesting property. If you perform all of the operations specified by the left-hand side of the equation–that is, take the second derivative of the wave function and then add the product of the potential energy times the wave function–the result is just a simple number multiplied by the wave function.
In mathematician’s parlance, whenever a differential operation on a wave function Ψ results in a simple number multiplied by Ψ, then the wave function is classified as an “Eigenfunction.” And, that simple number is called the Eigenvalue that goes along with that Eigenfunction.
Remember that energy is the name of the game. So it shouldn’t surprise you that, in this formulation of quantum mechanics, the Eigenvalues correspond to the total energies of the particle. Each Eigenfunction is therefore a particle’s wave function, and each Eigenvalue is the energy that goes with it.
ATOM TRAP
While the Eigenfunctions are unique and distinguishable wave functions, the energies associated with each Eigenfunction may not be unique. It is perfectly allowable for two different Eigenfunctions to have the same Eigenvalue. When this happens in quantum physics, we call it “degeneracy.” This occurs frequently when we use the Schroedinger equation to describe 3-dimensional problems, a situation we’ll soon encounter.
Remember that the Eigenfunctions and Eigenvalues are determined by the potential energy function that goes with a particular situation and the boundary conditions that force the math to conform to reality. The beauty of all this is that we finally have an explanation for the allowed states of electrons in atoms.
The experimenters had long observed that electrons could only exist in certain discrete states in the atom. These states will correspond exactly to the Eigenfunctions you get when the atom’s potential energy is put into Schroedinger equation. What’s more, those states have exactly the correct energies (Eigenvalues) that showed themselves as lines in atomic absorption and emission spectra decades before Schroedinger formulated his equation. The allowed atomic “orbitals” and quantized energy levels come naturally out of one guiding equation.
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