The Quantum Universe

by Brian Cox

  That looks like a mess and not much like we are within one page of hitting the jackpot. The key point is to notice that this is expressing a relationship between the mass of the star and its radius – a concrete relation between the two is within touching distance (or desperate grasping distance, depending on how well you handled the mathematics). After substituting in for the average density of the star (i.e. ) this messy equation can be rearranged to read

  Now λ only depends upon the dimensionless quantities a, b, f, g and h, which means that it does not depend upon the quantities that describe the star as a whole, M and R, and this means that it must take on the same value for all white dwarf stars.

  If you are worrying what would happen if we were to change a and/or b (which means changing the locations and/or size of our little cube) then you have missed the power of this argument. Taken at face value, it certainly looks like changing a and b will change λ so that we will get a different answer for RM1/3. But that is impossible, because we know that RM1/3 is something that depends on the star and not on the specific properties of a little cube that we might or might not care to dream up. This means that any variation in a or b must be compensated for by corresponding changes in f, g and h.

  Equation (5) says, quite specifically, that white dwarves can exist. It says that because we’ve been successfully able to balance the gravity–pressure equation (equation (1)). That is not a trivial thing – because it might have been possible that the equation could not be satisfied for any combination of M and R. Equation (5) also makes the prediction that the quantity RM1/3 must be a constant. In other words, if we look up into the sky and measure the radius and the mass of white dwarves, we should find that the radius multiplied by the cube root of the mass will give the same number for every white dwarf. That is a bold prediction.

  The argument that we just presented can be improved upon because it is possible to calculate exactly what the value of λ should be, but to do that we would need to solve a second-order differential equation in the density, and that is a mathematical bridge too far for this book. Remember, λ is a pure number: it simply ‘is what it is’ and we can, with a little higher-level maths, compute it. The fact that we did not actually work it out here should not detract at all from our achievements: we have proven that white dwarf stars can exist and we have managed to make a prediction relating their mass and radius. After calculating λ (which can be done on a home computer), and after substituting in the values for κ and G, the prediction is that

  which is equal to 1.1 × 1017 kg1/3m for cores of pure helium, carbon or oxygen (Z/A = 1/2). For iron cores, Z/A = 26/56 and the 1.1 reduces slightly to 1.0. We trawled the academic literature and collected together the data on the masses and radii of sixteen white dwarf stars sprinkled about the Milky Way, our galactic backyard. For each we computed the value of RM1/3 and the result is that astronomical observations reveal RM1/3 ≈ 0.9 × 1017 kg1/3m. The agreement between the observations and theory is thrilling – we have succeeded in using the Pauli Exclusion Principle, the Heisenberg Uncertainty Principle and Newton’s law of gravity to predict the mass–radius relationship of white dwarf stars.

  There is, of course, some uncertainty on these numbers (the theory value of 1.0 or 1.1 and the observational number equal to 0.9). A proper scientific analysis would now start talking about just how likely it is that the theory and experiment are in agreement, but for our purposes that level of analysis is unnecessary because the agreement is already staggeringly good. It is quite fantastic that we have managed to figure all this out to an accuracy of something like 10%, and is compelling evidence that we have a decent understanding of stars and of quantum mechanics.

  Professional physicists and astronomers would not leave things here. They would be keen to test the theoretical understanding in as much detail as possible, and to do that means improving on the description we presented in this chapter. In particular, an improved analysis would take into account that the temperature of the star does play some role in its structure. Furthermore, the sea of electrons is swarming around in the presence of positively charged atomic nuclei and, in our calculation, we totally ignored the interactions between the electrons and the nuclei (and between electrons and electrons). We neglected these things because we claimed that they would produce fairly small corrections to our simpler treatment. That claim is supported by more detailed calculations and it is why our simple treatment agrees so well with the data.

  We have obviously learnt an awful lot already: we have established that the electron pressure is capable of supporting a white dwarf star and we have managed to predict with some precision how the radius of the star changes if we add or remove mass from the star. Unlike ‘ordinary’ stars that are eagerly burning fuel, notice that white dwarf stars have the feature that adding mass to a star makes it smaller. This happens because the extra stuff we add goes into increasing the star’s gravity, and that makes it contract. Taken at face value the relationship expressed in equation (5) seems to imply that we would need to add an infinite amount of mass before the star shrinks to no size at all. But this isn’t what happens. The important thing, as we mentioned at the beginning of the chapter, is that we eventually move into the regime where the electrons are so tightly packed that Einstein’s Theory of Special Relativity becomes important because the speed of the electrons starts to approach the speed of light. The impact on our calculation is that we have to stop using Newton’s laws of motion, and replace them with Einstein’s laws. This, as we shall now see, makes all the difference.

  What we’re about to find is that as the star gets more massive, the pressure exerted by the electrons will no longer be proportional to the density raised to the power ; instead, the pressure increases less quickly with density. We will do the calculation in a moment, but straight away we can see that this could have catastrophic consequences for the star. It means that when we add mass, there will be the usual increase in gravity but a smaller increase in pressure. The star’s fate hinges on just how much ‘less quickly’ the pressure varies with density when the electrons are moving fast. Clearly it is time to figure out what the pressure of a ‘relativistic’ electron gas is.

  Fortunately, we do not need to wheel in the heavy machinery of Einstein’s theory because the calculation of the pressure in a gas of electrons moving close to light speed follows almost exactly the same reasoning as that we just presented for a gas of ‘slow-moving’ electrons. The key difference is that we can no longer write that the momentum p = mv, because this is not correct any more. What is correct, though, is that the force exerted by the electrons is still equal to the rate of change of their momentum. Previously, we deduced that a fleet of electrons bouncing off a mirror exerts a pressure P = 2mv × (nv). For the relativistic case, we can write the same expression, but providing that we replace mv by the momentum, p. We are also assuming that the speed of the electrons is close to the speed of light, so we can replace v with c. Finally, we still have to divide by 6 to get the pressure in the star. This means that we can write that the pressure for the relativistic gas as P = 2p × nc/6 = pnc/3. Just as before, we can now go ahead and use Heisenberg’s Uncertainty Principle to say that the typical momentum of the confined electrons is h(n/2)1/3 and so

  Again we can compare this to the exact answer, which is

  Finally, we can follow the same methodology as before to express the pressure in terms of the mass density within the star and derive the alternative to equation (4):

  where . As promised, the pressure increases less quickly as the density increases than it does for the non-relativistic case. Specifically, the density increases with a power of rather than . The reason for this slower variation can be traced back to the fact that the electrons cannot travel faster than the speed of light. This means that the ‘flux’ factor, nv, which we used to compute the pressure saturates at nc and the gas is not capable of delivering the electrons to the mirror (or face of the cube) at a sufficient rate to maintain the ρ5/3 behavio
ur.

  We can now explore the implications of this change because we can go through the same argument as in the non-relativistic case to derive the counterpart to equation (5):

  This is a very important result because, unlike equation (5), it does not have any dependence upon the radius of the star. The equation is telling us that this kind of star, packed with light-speed electrons, can only have a very specific value of its mass. Substituting in for κ′ from the previous paragraph gives us the prediction that

  This is exactly the result we advertised right at the start of this chapter for the maximum mass that a white dwarf star can possibly have. We are very close to reproducing Chandrasekhar’s result. All that remains to understand is why this special value is the maximum possible mass.

  We have learnt that for white dwarf stars that are not too massive, the radius is not too small and the electrons are not too squashed. They therefore do not quantum jiggle to excess and their speeds 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

  Published by the Penguin Group

  Penguin Books Ltd, 80 Strand, London WC2R 0RL, England

  Penguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014, USA

  Penguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario, Canada M4P 2Y3 (a division of Pearson Canada Inc.)

  Penguin Ireland, 25 St Stephen’s Green, Dublin 2, Ireland (a division of Penguin Books Ltd)

  Penguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of Pearson Australia Group Pty Ltd)

  Penguin Books India Pvt Ltd, 11 Community Centre, Panchsheel Park, New Delhi – 110 017, India

  Penguin Group (NZ), 67 Apollo Drive, Rosedale, Auckland 0632, New Zealand

  (a division of Pearson New Zealand Ltd)

  Penguin Books (South Africa) (Pty) Ltd, 24 Sturdee Avenue, Rosebank 2196, South Africa

  Penguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England

  www.penguin.com

  First published 2011

  Copyright © Brian Cox and Jeff Forshaw, 2011

  The moral right of the authors has been asserted

  Thanks to Paul Hetherington for recommending the front-cover font: Lÿon, designed by Radim Pesko and Karl Nawrot

  Cover art-directed by Peter Saville

  Cover photograph by Tina Negus

  All rights reserved.

  Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without the prior written permission of both the copyright owner and the above publisher of this book

  ISBN: 978-0-14-196803-2

  1. Unless of course you are reading an electronic version of the book, in which case you will need to exercise your imagination.

  1. But not so ridiculous when you consider that an oft-used unit of power, even to this day, is the ‘horsepower’.

  2. Once upon a time, televisions operated using this idea. A stream of electrons generated by a hot wire was gathered, focused into a beam and accelerated by a magnetic field towards a screen that glowed when the electrons hit it.

  1. For those who are familiar with mathematics, just exchange the words as follows: ‘clock’ for ‘complex number’, ‘size of the clock’ for ‘modulus of the complex number’ and ‘the direction of the hour-hand’ for ‘the phase’. The rule for adding clocks is nothing more than the rule for adding complex numbers.

  1. Or aesthetic appeal, depending on your point of view.

  2. If you are having trouble with that last sentence try replacing the word ‘clock’ with ‘wave’.

  3. The kinetic energy is equal to mv2/2 and the potential energy is mgh when the ball is a height h above the ground. g is the rate at which all objects accelerate in the vicinity of the Earth. The action is their difference integrated between the times associated with the two points on the path.

  4. Wikipedia describes ‘tripe’ as ‘a type of edible offal from the stomachs of various farm animals’, but it is colloquially used to mean ‘nonsense’. Either definition is appropriate here.

  5. Shrinking all clocks by the same amount is strictly only true provided that we are ignoring the effects of Einstein’s Special Theory of Relativity. Otherwise, some of the clocks get shrunk more than others. We shan’t need to worry about this.

  6. For a particle of mass m that hops a distance x in a time t, the action is ½m(x/t)2t if the particle travels in a straight line at constant speed. But this does not mean the quantum particle travels from place to place in straight lines. The
clock-winding rule is obtained by associating a clock with each possible path the particle can take between two points and it is an accident that, after summing over all these paths, the result is equal to this simple result. For example, the clock-winding rule is not this simple if we include corrections to ensure consistency with Einstein’s Theory of Special Relativity.

  7. A sand grain typically has a mass around 1 microgram, which is a millionth of a kilogram.

  8. There is a chance that the particle travels even farther than the ‘extreme’ case marked out by the large blob in the figure but, as we have shown, the clocks tend to cancel out for such scenarios.

  1. You might like to check this explicitly for yourself.

  2. ‘Diffraction’ is a word used to describe a particular type of interference, and it is characteristic of waves.

  3. Of course if d is very large then one might wonder how we can even measure the momentum. That concern is sidestepped by ensuring that no matter how big d is, L is much bigger than it.

  4. Recall that when we draw pictures of waves, they are really a convenient way of picturing what the projections of the clock hands in the 12 o’clock direction are.

  5. This way of arriving at the Uncertainty Principle did, however, rely on the de Broglie equation in order to link the wavelength of a clock wave to its momentum.

  6. In the jargon, the momentum space wavefunctions that correspond to particles with definite momentum are known as momentum eigenstates, after the German word eigen, meaning ‘characteristic’.