Warped Passages - Lisa Randall [70]
You could continue along these lines. If you were to perform a measurement for 100 seconds, you could measure frequency to an accuracy of water dripping once per 100 seconds. If you were to measure water dripping for 1,000 seconds, you could measure frequency with an accuracy of once per 1,000 seconds. In all these cases the product of the time interval over which you performed your measurement and the accuracy with which you measure frequency is about 1.* The longer time required for a more accurate measurement of frequency is at the heart of the time-energy uncertainty principle. You can measure frequency more accurately, but to do so you would have to measure for longer. The product of time and the uncertainty in frequency is always about 1.*
To complete the derivation of our simple uncertainty principle, if you had a sufficiently simple quantum mechanical system—a single photon, for example—its energy would be equal to Planck’s constant, h, times frequency. For such an object, the product of the time interval over which you measure energy and the error in energy would always exceed h. You could measure its energy as precisely as you like, but your experiment would then have to run for a correspondingly longer time. This is the same uncertainty principle we just derived; the added twist is only the quantization relation that relates energy to frequency.
Two Important Energy Values and What the Uncertainty Principle Tells Us About Them
That almost completes our introduction to the fundamentals of quantum mechanics. This and the following section review two remaining elements of quantum mechanics that we will use later on.
This section, which does not involve any new physical principles, presents one important application of the uncertainty principle and special relativity. It explores the relationships between two important energies and the smallest length scales of the physical processes to which particles with those energies could be sensitive—relationships that particle physicists use all the time. The following section will introduce spin, bosons, and fermions—notions that will appear in the next chapter, about the Standard Model of particle physics, and also later on, when we consider supersymmetry.
The position-momentum uncertainty principle says the product of uncertainties in position and in momentum must exceed Planck’s constant. It tells us that anything—whether it is a light beam, a particle, or any other object or system you can think of that could be sensitive to physical processes occurring over short distances—must involve a large range of momenta (since the momentum must be very uncertain). In particular, any object that is sensitive to those physical processes must involve very high momenta. According to special relativity, when momenta are high, so are energies. Combining these two facts tells us that the only way to explore short distances is with high energies.
Another way of explaining this is to say that we need high energies to explore short distances because only particles whose wavefunctions vary over small scales will be affected by short-distance physical processes. Just as Vermeer could not have executed his paintings with a two-inch-wide brush, and just as you can’t see fine detail with blurry vision, particles cannot be sensitive to short-distance physical processes unless their wavefunction varies over only small scales. But according to de Broglie, particles whose wavefunction involves short wavelengths also have high momenta. De Broglie said that the wavelength of a particle-wave is inversely proportional to its momentum. Therefore de Broglie would also have us conclude that you need high momenta, and hence high energies, to be sensitive to the physics of short distances.
This has important ramifications for particle physics. Only high-energy particles feel the effects of short-distance physical processes. We’ll see in two specific cases just how high I mean.
Particle physicists often measure energy in multiples of an electronvolt, which