The Correspondence Principle
How Bohr Managed to Summarise the limits of Universal Laws into a Single Mathematical Proof?
I know what you're thinking. We all know that the Theory of Relativity is deemed as the theory when it comes to explaining all universal interactions - interactions occurring on a macro-scale. Of course, Einstein's Theory of Relativity is known for its elegant and remarkable picture of our entire Universe, a Universe that is an interplay between mass, energy, and spacetime. But we also know that Classical Physics, which includes Einstein's Theory of Relativity, isn't the only framework carrying the entire realm of Physics.
Since its introduction in 1926 by Erwin Schrodinger (yes, the cat-in-a-box guy!), the domain of Quantum Mechanics has hugely impacted our understanding of the Universe pertaining to its micro-scale - or the subatomic world. Principles of Quantum Mechanics have shown that not only is the subatomic world nothing like the macro universe, but its predictions and observations show entirely non-intuitional results!
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| Credits: Stanford University |
Even with all of its weirdness, Quantum physics has influenced our daily life through numerous innovations and technology, proving that both Classical and Quantum faces of physics, despite their obvious incompatibility with each other, are unveiling the mysteries of the Universe.
In fact, Neils Bohr, one of the pioneers of Quantum Physics, made this point clear through his Great Debate with Einstein in 1927. Einstein objected to the probabilistic nature of observing quantum systems, arguing that Quantum Mechanics is not a worthy framework to be considered in Modern physics. Bohr, on the other hand, defended the probabilistic nature of the subatomic world, and after multitudes of arguments and thought experiments, ended up winning the debate!
But what is surprising is that despite the debates and disagreements with each other's opinion on the 'right theory in physics', both Einstein and Bohr significantly contributed to their rival teams. Einstein, because of his discovery of the photoelectric effect (which won him the Nobel Prize in 1921), is considered one of the founding fathers of Quantum Mechanics! And as for Bohr, he just casually came up with a mathematical proof that showed that the Universe can only be accurately explained by applying both Quantum Physics and Classical Physics!
In other words, Bohr came up with a principle that showed that quantum principles will reproduce the original classical principles when applied to larger orbits and energies (when quantum number, n → ∞). This is known as the Correspondence Principle, and it is the most beautiful representation of the connection between the Classical and Quantum worlds.
Now, Bohr validates this approximation of quantum mechanical laws to that of classical laws by working out the mathematics based on his famous model of the Hydrogen atom. Bohr made his point by simply coming up with a relation between the frequency of revolution of an electron orbiting its nucleus and the frequency of radiation emitted by an electron when jumping from a higher to a lower energy level. Now I know that sentence doesn't seem easy to process, but let's break it down bit by bit to make it easier to grasp.
We know that in classical electromagnetic theory, the frequency of light emitted by an electron orbiting the nucleus is equal to the frequency of its revolution. This notion that orbiting electrons produce electromagnetic radiation was first brought forward by Rutherford. We also know that according to quantum theory, an electron emits electromagnetic radiation when transitioning from higher energy (excited) state to a lower energy (ground) state.
So, going back to the previous statement, Bohr managed to show the connection between these two properties derived from entirely different domains - one being a classical property of an electron and the other being a quantum property of an electron. To be more clear, Bohr essentially came up with a way to show that the frequency of revolution of an electron that we obtain classically is related to the frequency of the emitted radiation by the transitioning electron, which we obtained using quantum principles. The result of the Correspondence Principle is that at larger quantum numbers (at n >>), the frequency of the radiation emitted by the transitioning electron will be equal to the frequency of the revolution of the electron. This simply means that at large n values, quantum mechanics will produce classical results!
This principle explains why physical systems in the macro universe (larger n values), do not exhibit quantum behaviours like quantum superposition and quantum tunnelling. This is also the reason why we cannot apply quantum mechanical principles on a macro scale, as the classical effects will dominate in such a case.
Now of course, this does seem like another one of those typical proofs in Physics when you casually think about it, especially if you take a look at the actual mathematical proof (link can be found at the end of the post). But this principle gives a much more deeper appreciation of where our age-old notions of the Universe stand today, and how it has evolved over thousands of years. Truly, humankind is limitless when it comes to unravelling the secrets of the Universe!
With that being said, stay tuned for an exclusive post on Quantum Mechanics core - describing wavefunctions and some insights into its amplitude and properties.
The Correspondence Principle - Arthur Beiser (Chapter - 4.6, Pg - 138)
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