Warped Passages - Lisa Randall [69]
In this derivation, we’ll focus on time-energy uncertainty, which is a little easier to explain and understand. The time-energy uncertainty principle relates the uncertainty in energy (and hence, according to Planck’s assumption, frequency) to the time interval that is characteristic of the rate of change of the system. That is, the product of the energy uncertainty and the characteristic time for the system to change will always be greater than Planck’s constant, h.
A physical realization of the time-energy uncertainty principle happens, for example, when you turn on a light switch and hear static from a nearby radio. Turning on the light switch generates a large range of radio frequencies. That’s because the amount of electricity going through the wire changed very rapidly, so the range of energy (and hence frequency) must be large. Your radio picks it up as static.
To understand the uncertainty principle’s origin, let’s now consider a very different example—a leaky faucet.* We will show that you need a long-lasting measurement to accurately determine the rate at which the faucet drips, which we will see is closely analogous to the uncertainty principle’s claim. A faucet and the water passing through it, which involve many atoms, is too complex a system to exhibit observable quantum mechanical effects—they are overwhelmed by classical processes. It is nonetheless true that you need longer measurements to make more accurate frequency determinations—and that is the core of the uncertainty principle. A quantum mechanical system would take this interdependence a step further because for a carefully prepared quantum mechanical system, energy and frequency are related. So for a quantum mechanical system, a relation between frequency uncertainty and the length of time of a measurement (like the one we are about to see) therefore translates into the true uncertainty relation between energy and time.
Suppose that water is dripping at a rate of about once per second. How well could you measure the rate if your stopwatch had one-second accuracy—that is, it could be off by at most one second? If you were to wait one second, and saw a single drip, you might think that you could conclude that the faucet drips once per second.
However, because your stopwatch could be off by as much as one second, your observation wouldn’t tell you precisely how long it took for the faucet to drip. If your watch ticked once, the time might have been a little more than one second, or it might have been nearly two. At what time, between one and two seconds, should you say the faucet dripped? Without a better stopwatch or a longer measurement, there would be no preferred answer. With the watch you have, you can conclude only that the drops fall somewhere in the range between one per second and one per two seconds. If you said that the faucet drips once per second, you could have essentially 100% error in your measurement. That is, you could be off by as much as a factor of two.
But suppose that instead you waited 10 seconds while performing your measurement. Then, 10 drops of water would have fallen during the time it took your watch to click 10 times. With your crude stopwatch with only 1-second accuracy, all you could really infer is that the time it took for 10 drips was somewhere between 10 and 11 seconds. Your measurement, which would again say that the drips fall approximately once per second, would now have an error of only 10%. That’s because by waiting 10 seconds, you could measure frequency to within 1/10 of a second. Notice that the product of the time for your measurement (10 seconds) and the uncertainty in frequency (10 %, or 0. 1) is was roughly 1. Notice also that the product of uncertainty in frequency and time for the measurement in the first example, which had more error in the frequency measurement (100 %) but took place over