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are small compared to the speed of light. For these stars, we have seen that they are stable with a mass–radius relationship of the form RM1/3 = constant. Now imagine adding more mass to the star. The mass–radius relation informs us that the star shrinks and, as a result, the electrons are more compressed and that means they jiggle faster. Add yet more mass and the star shrinks some more. Adding mass therefore increases the speed of the electrons until, eventually, they are travelling at speeds comparable with the speed of light. At the same time, the pressure will slowly change from and in the latter case, the star is only stable at one particular value of the mass. If the mass is increased beyond this specific value then the right-hand side of becomes larger than the left-hand side and the equation is unbalanced. This means that the electron pressure (which resides on the left-hand side of the equation) is insufficient to balance the inward pull of gravity (which resides on the right-hand side) and the star must necessarily collapse.

If we were more careful with our treatment of the electron momentum and had taken the trouble to wheel in the advanced mathematics to compute the missing numbers (again a minor task for a personal computer), we could make a precise prediction for the maximum mass of a white dwarf star. It is

where we have re-expressed the bundle of physical constants in terms of the mass of our Sun . Notice, by the way, that all the extra hard work that we have not done simply returns the constant of proportionality, which has a value of 0.2. This equation delivers the sought-after Chandrasekhar limit: 1.4 solar masses for Z/A = ½

This really is the end of our journey. The calculation in this chapter has been at a higher mathematical level than the rest of the book but it is, in our view, one of the most spectacular demonstrations of the sheer power of modern physics. To be sure, it is not a ‘useful’ thing, but it is surely one of the great triumphs of the human mind. We used relativity, quantum mechanics and careful mathematical reasoning to calculate correctly the maximum size of a blob of matter that can be supported against gravity by the Exclusion Principle. This means that the science is right; that quantum mechanics, no matter how strange it might seem, is a theory that describes the real world. And that is a good way to end.

Further Reading

We used many books in the preparation of this book, but some deserve special mention and are highly recommended.

For the history of quantum mechanics, the definitive sources are two superb books by Abraham Pais: Inward Bound and Subtle Is the Lord … Both are quite technical but they are unrivalled in historical detail.

Richard Feynman’s book QED: The Strange Theory of Light and Matter is at a similar level to this book and is more focused, as the title suggests, on the theory of quantum electrodynamics. It is a joy to read, like most of Feynman’s writings.

For those in search of more detail, the very best book on the fundamentals of quantum mechanics is, in our view, still Paul Dirac’s book The Principles of Quantum Mechanics. A high level of mathematical ability is needed to tackle this one.

Online, we should like to recommend two lecture courses that are available on iTunes University: Leonard Susskind’s ‘Modern Physics: The Theoretical Minimum – Quantum Mechanics’ and James Binney’s more advanced ‘Quantum Mechanics’ from the University of Oxford. Both require a reasonable mathematical background.

ALLEN LANE

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