I should some words about quantum mechanics to begin with as that topic will permeate everything from now on.
Quantum mechanics is held as being inscrutable, and one of the best theories we have. It is claimed to produce extremely precise and accurate predictions. And it is true!
From this, it is tempting to conclude that the story has ended. So what if it is inscrutable? Surely this is just because it is mathematically complex? Whoever said the universe had to be simple? Isn’t it enough that it is predictable?
I think that this claim is false: the story of quantum mechanics has not ended.
In fact, quantum mechanics in its standard form is fundamentally flawed. In certain circumstances, as you will see, it refuses to make any predictions at all!
Quantum mechanics in its standard form is actually two theories.
The first part is the Schrodinger mechanics which describes the system as an evolving correlated many particle wave. Apart from the fact that a particle either has a momentum or a position, not both, the correspondence to classical mechanics is clear.
The second part is the infamous postulate which states that after a measurement the system discontinususly changes from one wave to another. The latter wave is by definition one of the normalized eigenstates of the measurement operator; and the probability that one obtains this state depends on the “overlap” of the initial and final wave functions.
The second part is nonsense!
How does the collapse occur? How long does it take? What is the phase of the final wave? Is that random as well? Standard quantum mechanics forbids us to even ask the question!
In a later post I will explain why quantum mechanics in it’s current form must be a statistical theory. And in any statistical theory there is an inherent unexplained component. For example, when one takes statistics on the heights of people in a population, the reason why those people actually have a height to be measured is not of interest. We just collect statistics. Quantum mechanics makes such statistical predictions as you can see.
There is a school of thought, the Copenhagen interpretation, which argues that this is all we can ever know. In other words, it is meaningless to ask what happens during the measurement-collapse event. This interpretation goes too far.
Here is why:
The problem is, the Copenhagen interpretation, when applied to doing statistics on the heights of people, would encourage us not enquire into what height actually is. How did these people become tall or short? The Copenhagen interpretation says: “Sorry, I’m only allowed to tell you the statistical result!”. Now, perhaps in the case of the quantum world, that may be the case, but is is nevertheless unsatisfactory.
Besides, if it is meaningless to question what happens during a quantum measurement event, it must also be equally meaningless to make assertions about the nature of that event. But the Copenhagen school makes just such an assertion! Namely: that we are not allowed to question how the collapse-event occurs. The event is supposed to be incognito. Clearly then, making such an assertion is logically inconsistent by it’s own standards!
Luckily much progress has been made in investigating these issues. The theory of quantum decoherence is currently very fashionable and compelling. One of it’s champions, Zurek, writes elegantly on the topic but for my taste, not quite convincingly. On the other hand others such as Deutch make an equally compelling case for the “many worlds” interpretation.
What has all this to do with quantum chemistry?
Quite a lot, actually.
I will give two examples.
There is a tendency in this field to depend very much on the statements made by Dirac to the effect that “all of chemistry is now explained” by quantum mechanics. The arguments above show that this is clearly false. It may be the case for model systems which are highly decoupled from their environments. However, it is in chemistry where systems become large and complex, and it is in precisely these cases that the notions of system-separability become critical. Not everything can be regarded usefully as an isolated system. To give one example, consider the case of electrical conductivity through a single molecule.
Second, quantum chemists often emphasis on Hermitian operators. The claim is that “all observables must be Hermitian”. Certainly, Hermitian operators have real eigenvalues, and the results of measurements should be real. And a Hermitian operator ensures that the time development of the wavefunction is unitary, leading to a normalized wave function for all time. However, many measurements are conveniently described by complex operators. For example, in the theory of scattering there are situations where particle probability is not conserved with time — corresponding to particle absorption, decay, or destruction. There are other experiments in diffraction where it is convenient if one assumes the phase as well as the amplitude of a wave is measured; these experiments too are conveniently regarded in terms of complex operators.
All this is to indicate to you: always be on guard about grand statements like “everything is known’’, or “this is exact”. Implicit in these is always the model. Always.
At this point I should present a quite remarkable exchange between Einstein and Heisenberg, which the latter included in his memoirs. It clearly illustrates what I have been trying to say. But it will have to wait for another post.
The bright side
I don’t want to give you the wrong impression.
Quantum mechanics does produce great results. Hence, it is well worth developing a feeling for it. If you don’t think that is true, you should keep reading papers, keep doing your own calculations. The best way is if you do the work yourself, it becomes much more convincing that way. Hence this blog.
From my experience, it is really is quite astounding how the mathematics of quantum mechanics can be used to produce the same numbers that an experimentalist does: each of us works laboriously on seemingly quite divergent directions and at the end of the day we invariably get the same result.
Now my old supervisor, Handy, used to say
Quantum chemistry is the last field where, as the system becomes bigger, you get reproducible results.
By this he meant that calculations on systems which are any bigger invariably require statistical mechanical techniques, and these have an intrinsic irreproducibility due to random random sampling.
If you accept this, then it follows that quantum chemistry is the last field where concerns about separability of system and environment can be numerically probed and experimentally tested. We invariably get the same result as the experimentalist, but do we always hae to?
Are quantum chemists actually looking for cases where the results might differ? I don’t know. Why don’t you try?