Thanks to the double slit experiment, we already know one of the principles of quantum mechanics, namely that we are able to calculate the correct probability amplitudes for a system by describing this system as a set of partial amplitudes weighted by coefficients. To begin with, we want to consider a system with only two states with which, nevertheless, we are able to explain many phenomena in chemistry and spectroscopy. We imagine benzene with the following pair of electronic structures.
Benzene = | ![]() |
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This may seem hazardous as we know neither the amplitudes
|1> and |2> nor the respective energy of benzene in these two electronic configurations. Nevertheless, we pretend having an energy and introduce a more or less obvious aspect: Due to symmetry, both configurations have identical energies which will be denoted as Eo. Another quick assumption could be that even the coefficients a1 and a2 are the same, but we have to be aware that the probability density is proportional to the square of ψ. Thus, the sign of the coefficients remains questionable and, at best, we are able to state |a1|=|a2|. But additional mathematical aspects pave the way to a solution.
For a start, although the Hamilton operator H is undetermined, we have Schrödinger's equation for benzene.
Within this equation, the wave function ΨBenzene is replaced by the linear combination introduced above.
We multiply the equation with <1| and with <2| and apply the distributive law subsequently. As we know the states <1| and <2| to be orthonormal, the respective values of one and zero for the terms <1|1> and
Multiplied with <1| : | ![]() |
Multiplied with <2| : | ![]() |
By combining both equations, we eliminate the fraction a1/a2 and find the energy states EI and EII.
We draw the conclusion that there are two stable states, one of them of magnitude W below, the other of magnitude W above E0. Usually, EI is the energy of lower, EII the energy of the higher state (a convention that would favour reversing the ± sign, but we won't care for that here). In case |1> and |2>> do not at all interfere with each other, EI , EII and Eo are all the same.
Instead of W, we now write -ΔE. For the energies of the two stationary states |I> and |II> we then recieve
At last, we want to characterize the shape of electronic configuration |I> and |II> Therefore, we insert the solutions for E into the expression found for a1/a2 derived above. This establishes a1 = ±a2 as mathematical link between the two coefficients, a relation already expected due to general considerations. If we finally introduce the conditions <I|I> = 1, and <II|II> = 1, i.e. normalize the established linear combinations, we finally get