The state of a quantum system is labeled by so-called ‘quantum numbers’. We revisit this concept:
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The O2-molecule has 16 electrons and 2 nuclei. Without making the Born-Oppenheimer approximation, this is a quantum system with 18 particles. With 3 spatial degrees of freedom and one spin degree of freedom per particle, this makes (16+2)x(3+1)=72 quantum numbers.
If the Born-Oppenheimer approximation is made, then the nuclei are frozen at a given position. Only the electrons are left as members of the quantum system. There are now 16 particles with 4 degrees of freedom each, which requires 16×4=64 quantum numbers.
Consider the O2-molecule. How many quantum numbers would you need to label the many-body wave function of this entire molecule? Give your answers once with and once without the Born-Oppenheimer approximation.
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