Warped Passages - Lisa Randall [68]
Heisenberg hated the external upheaval that surrounded him in his youth; he would have preferred a return to “the principles of Prussian life, the subordination of individual ambition to the common cause, modesty in private life, honesty and incorruptibility, gallantry and punctuality.”† Nonetheless, with the uncertainty principle Heisenberg irrevocably changed people’s worldview. Perhaps the tumultuous era in which Heisenberg lived gave him a revolutionary approach to science, if not to politics.‡ In any case, I find it a little ironic that the author of the uncertainty principle was a man of such conflicting dispositions.
The uncertainty principle says that certain pairs of quantities can never be measured accurately at the same time. This was a major departure from classical physics, which assumes that, at least in principle, you can measure all the characteristics of a physical system—position and momentum, for example—as accurately as you’d like.
The particular pairs are those for which it matters which one you measure first. For example, if you were to measure position and then momentum (the quantity which gives both speed and direction), you wouldn’t get the same result as if you first measured momentum and then position. This would not be the case in classical physics, and is certainly not what we are used to. The order of measurements matters only in quantum mechanics. And the uncertainty principle says that for two quantities where the order of measurement matters, the product of the uncertainties of the two will always be greater than a fundamental constant, namely Planck’s constant, h, which is 6.582 × 10-25 GeV second for those who want to know.*12 If you insist on knowing position very accurately, you cannot know momentum with a similar accuracy, and vice versa. No matter how precise your measuring instruments and no matter how many times you try, you can never simultaneously measure both quantities to very high accuracy.
The appearance of Planck’s constant in the uncertainty principle makes a good deal of sense. Planck’s constant is a quantity that arises only with quantum mechanics. Recall that, according to quantum mechanics, the quanta of energy of a particle with a particular frequency is Planck’s constant times that frequency. If classical physics ruled the world, Planck’s constant would be zero and there would be no fundamental quantum.
But in the true quantum mechanical description of the world, Planck’s constant is a fixed, nonzero quantity. And that number tells us about uncertainty. In principle, any individual quantity can be accurately known. Sometimes physicists refer to the collapse of the wavefunction to specify that something has been accurately measured and therefore takes a precise value. The word “collapse” refers to the shape of the wavefunction, which is no longer spread out but takes a nonzero value at one particular place, since the probability of measuring any other value afterwards is zero. In this case—when one quantity is measured precisely—the uncertainty principle would tell you that after the measurement, you can know nothing at all about the other quantity that is paired with the measured quantity in the uncertainty principle. You would have infinite uncertainty in the value of that other quantity. Of course, had you first measured the second quantity, the first quantity would be the one you didn’t know. Either way, the more accurately you know one of the quantities, the less precise the measurement of the other has to be.
I won’t go into the detailed derivation of the uncertainty principle in this book, but I’ll nonetheless try to