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The title of this post might suggest we are going to solve problems in quantum mechanics, like calculating the Bohr radius of the hydrogen atom. This is however left as an exercise to the reader, and I will give the solution straight away, which is a0 = 0.5292 Å. This post will be devoid of formulas, I promise.
Here I want to explain, in relatively simple terms where possible, problems in quantum mechanics itself. This may surprise you because this theory is very successful and brought us all kind of modern technology, like the very servers that host my blog and the device on which you read this post, so how could it be problematic? Still, these discussions started when quantum mechanics was a nascent discipline, and we still have not seen the end of it. There is no reason to be alarmed, as the theory would not be rudely set aside, nor would the quest for a quantum computer be abandoned, but the formalism is probably not yet in its final form.
Many scientists I know would not stop to consider potential flaws in its logic. I encourage students and professionals however to study the problems in the scientific theories we are taught, and topics in philosophy. For instance, I worked on an essay on the EPR paradox outlined below, when the issue became actual again due to the at the time recent work of Aspect in the early 1980's. Such studies give a deeper understanding of a scientific formalism and unresolved issues in their framework. Leading scientists challenge accepted formalisms, and further science significantly by resolving the tensions they recognise. That could be you, one day. Planck put forward energy quantisation since classical theory could not model black body radiation. Einstein conceived his theory on relativity because the electromagnetic behaviour of a moving body does not follow from combining Maxwell's and Newton's laws. Science, like a work of art, or a blog post for that manner, is never finished. Now, what is going on in quantum mechanics?
In a previous post I explained that in 1900, Planck derived the first good model for black body radiation. For this, he had to make the radical assumption that radiated energy could only be a discrete multiple of some tiny basic amount. Planck received the 1918 Nobel prize for this work. Einstein took this a step further by assuming that light itself is quantised as photons, which allowed him to explain the photoelectric effect in 1905, a feat that got him the 1921 Nobel prize. Nota bene, Einstein did not receive the Nobel for general relativity as you might presume, since that theory, well established now, was considered too controversial at the time.
The quantum of energy in Planck's model is minute, therefore in the classical, macroscopic world energy appears continuous for any practical purpose. It is at molecular or even smaller scale, that nature starts to behave in a manner that is unlike our everyday experience. In 1913 Bohr proposed a model, now obsolete, for a hydrogen atom in which electrons exist in quantised orbits. The absorption spectrum of hydrogen gas could now be explained from first principles, and with great accuracy at that. You see, when you analyse the colours in sunlight, you will notice that some very specific colours are missing giving a fingerprint of solar chemical composition, which is mainly hydrogen. What the nascent discipline lacked at that point was a consistent, unifying framework. That came when De Broglie proposed that not just photons, but all matter would have wave-like properties, whereafter Heisenberg developed matrix mechanics and Schrödinger devised wave mechanics. This framework is the quantum mechanics we chemists learn in university classes today.
Quantum mechanics has important philosophical implications. Newtonian mechanics or general relativity is objective and deterministic, meaning that phenomena manifest themselves in the same manner to everyone, and once you have all initial conditions then you know all about the system going forward. Quantum mechanics on the other hand is objective although indeterministic. In the 1920's, the view arose that you cannot know anything definite about a quantum system until you perform an, inherently irreversible, measurement. This view has been questioned as you can imagine - the term measurement is not well defined for instance, and how would a system know that it is being probed? Over the decades, other interpretations were formulated like the many-world interpretation that will come back later in this post. Even so, variants of the Copenhagen interpretation described above, are still a dominant conviction among scientists. Because of the different interpretations, other issues described in the next paragraphs, cannot be closed yet.
The particle-wave duality has a peculiar physical consequence at the quantum level, formulated by Heisenberg in 1927 as the uncertainty principle. In quantum mechanics, certain pairs of physical properties like position and momentum of a miniscule particle cannot both be known with certainty at the same time. All these revolutionary ideas emerged in a time when eminent minds as Einstein insisted that the universe is rational, and reality is certain in time and space. Considering quantum indeterminism, we have all heard Einstein's remark that God does not roll dice. He also allegedly asked whether the moon exists when nobody looks, referring to the uncertainty quantum theory implied. Einstein did not dismiss quantum mechanics, he was one of the founders as you will recall, but he did not consider it a final theory. That brings us to one tough question he brought forward together with two co-authors.
This well known debate already mentioned in the introduction is the Einstein-Podolsky-Rosen paradox, or EPR paradox, from 1935. It was a serious challenge to quantum mechanics that made headlines, and scientists have worked for decades to bring it to a conclusion. A strange consequence of quantum mechanics is that when two close, interacting quantum particles get separated far apart, one still instantaneously reacts to what happens to the other. This is called entanglement. According EPR, quantum mechanics would be incomplete because of this implied faster-than-light interaction. When you hear physicists say that quantum mechanics is a non-local theory, it is this action-at-a-distance they are talking about. Einstein et alii believed in locality, meaning that objects could only interact with what is near them, and devised the EPR paradox to make their point that quantum mechanics was correct though incomplete.
The non-locality issue lingered until Bell devised a theorem in 1964 that enabled deciding the matter experimentally. Laboratory work by Clauser in 1970 and later by Aspect in 1982 allowed for further validation of entanglement. Collected experimental data to this date, convincingly imply that one or more of the assumptions in the Bell theorem are false, by which entanglement appears to be a real phenomenon in the quantum world. However, which assumptions in the theorem that would be, differs per interpretation of quantum mechanics. Therefore, while many scientists now consider EPR disproven and quantum mechanics as a non-local theory, others abstain from a conclusion until we decide which interpretation of the theory is the correct one. Note however that entanglement is a quantum mechanical idiosyncrasy without practical application, since there is no way to have entangled particles do your bidding. This brings us to wave function collapse, another problematic concept.
Remember that in the Copenhagen interpretation, only upon measurement you will get definitive information about a quantum system. You will have heard of Schrödinger's cat trapped in a box with a contraption that could kill it by chance. In quantum mechanics, while the box remains closed the hapless animal is both alive and not so, until you open the box to find that it either is alive or has ceased to be. An unobserved system is thought to be in a superposition of all possible states. Upon measurement, this superposition collapses into a single state. This is where the microscopic quantum world is very different from the macroscopic world. An example up the alley of a chemist like me, is an unprobed hydrogen electron existing in all orbitals and spins at once. When you probe said electron, it reveals itself for example in the 2pz orbital with spin up.
Prominent minds like Penrose argue that Schrödinger's wave mechanics and the concept of collapse are fundamentally inconsistent. Note that this inconsistency depends on accepting a variant of the Copenhagen interpretation as being correct. A few years ago I wrote a post that mentioned the multiverse conjecture. Everett proposed a similar concept for quantum mechanics which is now called the many-world interpretation, in which possible quantum states do not overlap, but exist each in an own universe that has no interaction with other universes. Although it resolves the measurement problem and paradoxes like EPR, this is an untestable idea, as is the multiverse conjecture. Should we accept an interpretation that is essentially unscientific?
In the introduction I already mentioned that there is no reason for alarm. Quantum mechanics has turned our understanding of the microscopic world upside down, and we are still trying to get our collective minds around its consequences. This will in the end lead to an improved, even more successful, theory. One path, for example, is De Broglie-Bohm theory which is a deterministic though non-local variant of quantum mechanics. Alternatively, Penrose proposed that wave function collapse may be caused by gravity, not by observation. In technical terms this means that when spacetime curvature differences between states exceed a threshold, superposition destabilises. This idea elegantly blends Einstein's relativity and quantum mechanics. Would you want a worthy exercise in quantum mechanics after all, dear reader, then please unify its framework with general relativity. If you would succeed, you will be awarded the Nobel prize, no less.
Published in Essays. More on Education, Philosophy or Science.
© 1993-2026 J.M. van der Veer
jmvdveer@algol68genie.nl
