The Uncertainty Principle¶
Heisenberg introduced his famous uncertainty principle in 1927. 1 He based it on the following argument.
Suppose you want to measure the position of a particle. You might do it, for example, by shining light on it and detecting the reflected light. This experiment, like all real world experiments, has limited precision. It can only determine the particle’s position to an accuracy roughly equal to the wavelength of the light.
The interaction between the particle and the light perturbs the motion of the particle, imparting some momentum to it. We do not know the exact change in momentum (that would require knowing precise details of the interaction that cannot be determined from the experiment), but we do know its magnitude must be on the order of the amount of momentum carried by the light. That is inversely proportional to its wavelength. Combining these two results, Heisenberg obtained the following very approximate inequality.
where \(\Delta x\) is the uncertainty in our measurement of the position and \(\Delta p\) is the uncertainty in the momentum after completing the measurement. We could reduce \(\Delta x\) by using light with a shorter wavelength, but then we would disturb the particle more and increase the uncertainty in its momentum. Alternatively we could decrease \(\Delta p\) by using light with a longer wavelength, but then we would be able to measure the position less precisely.
This is a purely mechanistic argument based on the practical details of performing measurements. It also requires only classical physics. Nothing about it is specific to quantum mechanics.
The uncertainty principle was later reinterpreted as a statement about the quantum mechanical wavefunction. If a particle’s position is precisely known, its wavefunction is a delta function in position space. This corresponds to a plane wave in momentum space, meaning the momentum is completely unknown. Likewise, if the momentum is precisely known, the wavefunction is a delta function in momentum space and a plane wave in position space. If we want to know both position and momentum at the same time, we must choose a wavefunction that is localized but not precisely defined in both spaces. The best we can do turns out to be a Gaussian distribution, whose Fourier transform is also a Gaussian. This leads to the inequality
where \(\Delta x\) and \(\Delta p\) are now more precisely defined as being the standard deviations of the position space and momentum space representations of the wavefunction.
These two “uncertainty principles” seem entirely different from each other. They are based on entirely different arguments and relate to different phenomena (the practical limitations of the measurement process versus the intrinsic properties of the wavefunction). Yet they reach the same conclusion to within a small constant factor. Is this just a remarkable coincidence, or does it reflect a deeper connection?
If we view the wavefunction as a description of our knowledge about a system, the agreement is not surprising. It is only natural we would have settled on a description that reflects the actual limits of what we can know. On the other hand, if we interpret the wavefunction as a physical object rather than a mathematical abstraction invented by humans, the agreement is harder to explain. How does a purely classical description of the measurement process produce the correct result for an inherently non-classical phenomenon?
Although the two arguments lead to the same result, there is a critical difference between them. Heisenberg’s mechanistic argument applies only to initial value problems, not to boundary value problems. We can see this from a simple example.
Consider a single particle moving in free space. We first measure its position by reflecting light off it. Then we wait for a time interval \(t\), measure its position a second time, and find it has traveled a distance \(x\). We can conclude that between the two measurements, it must have been moving with velocity \(x/t\).
Of course, there is uncertainty in all of these values. The distance travelled is only measured to within some uncertainty \(\Delta x\) determined by the wavelength of the light. The time interval between the measurements can only be controlled to some precision \(\Delta t\). The ratio of distance to time therefore is also uncertain.
But we are not bound by the uncertainty principle. We can make \(\Delta x\) arbitrarily small by using shorter wavelength light. This does not increase the uncertainty in the velocity, because \(x/t\) is a direct measure of how quickly the particle was moving between the two measurements, after its momentum was disturbed by the first measurement and before it was disturbed by the second measurement. We also can make the uncertainty in the velocity smaller by increasing \(t\). Simply by choosing the wavelength of light and the time interval between the measurements, we can simultaneously know both position and momentum to arbitrary precision at every intermediate time between the two measurements.
Having determined the momentum through our two measurements, we might be tempted to extrapolate to earlier and later times. At this point we run into the uncertainty principle again. Each of the measurements alters the momentum in an unknown way. We know the velocity before the first measurement is different from \(x/t\), and after the second measurement it is different again, but we do not know the exact values. The more precisely we perform the position measurements, the more uncertain the momentum becomes.
If we instead view the wavefunction as a physical object rather than a mathematical abstraction, we are forced into a much more convoluted interpretation. Although we measured the particle to have traveled a distance \(x\) in time \(t\), we are not allowed to conclude it was traveling with velocity \(x/t\)! In this view, the first position measurement leads to a wavefunction that is localized in position and therefore uncertain in momentum. From that point, the uncertainty in position grows steadily. By the time we perform the second measurement, the wavefunction has come to be widely distributed. We measure the distance traveled to be \(x\), but we could just as easily have gotten a different result. Until we perform the second measurement and collapse the wavefunction, the particle’s position is highly uncertain. The particular position it happens to collapse to implies nothing about its momentum prior to the measurement.
In the same paper introducing the position-momentum uncertainty principle, Heisenberg also proposed a second uncertainty principle involving energy and time. He based it on a mechanistic argument similar to the other one. He considered a typical experiment used to measure energy and argued that the less time one spent performing the measurement, the less accurately one could determine the result. A simple argument led to the approximate inequality
where \(\Delta E\) is the uncertainty in the measured energy of a system and \(\Delta t\) is the amount of time taken to measure it.
This result is fundamentally different from the position-momentum uncertainty principle. “The time taken to perform a measurement” is not itself a measurable quantity. It is not something you can calculate from a wavefunction. It is a property of how you perform an experiment.
Just as the position-momentum uncertainty principle was later reinterpreted, the energy-time principle was also reinterpreted, but this time the meaning changed enormously in the process. Consider an eigenstate of the Hamiltonian. This is a state with precisely defined energy. Because it is an eigenstate, all measurable quantities are independent of time. They never change.
Now consider a superposition of states with similar but slightly different energies. This is not an eigenstate of the Hamiltonian, so measurable quantities do change with time. The wider the spread in energies, the more quickly they change. One can derive the inequality
where \(\Delta E\) now refers to the spread in energy of the states making up the superposition, and \(\Delta t\) describes the time scale over which any measurable quantity other than energy changes.
Beware of a common misunderstanding about the energy-time uncertainty principle. Many authors claim it means you can violate conservation of energy by an amount \(\Delta E\), as long as you only do it for a short time \(\Delta t\). This claim is entirely wrong! The expectation value of the Hamiltonian \(\left\langle \Psi | H | \Psi \right\rangle\) is conserved by the Schrödinger equation. That is what conservation of energy means in quantum mechanics. It is an exact law. It cannot be violated by even the smallest amount, no matter how short a time you do it for!
- 1
Heisenberg, W. “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik.” Zeitschrift für Physik 43: 172-198 (1927).