THE UNCERTAINTY PRINCIPLE

We have just seen that, for the probability interpretation of particle-waves to be a viable theory, there can be no way we can detect which slit the electron went through without destroying the two slit pattern. Also we have seen that if every particle and every force have a particle wave nature obeying the de Broglie relationship λ = h/p , then there is no way we can tell which slit the electron went through without destroying the two slit pattern. Both the particle-wave nature of matter, and the probability interpretation of particle waves, lead to a basic limitation on our ability to make experimental measurements. This basic limitation was discovered by Werner Heisenberg shortly before Schrödinger developed his wave equation for electrons. Heisenberg called this limitation the uncertainty principle.
When you cannot do something, when there is really no way to do something, physicists give the failure a name and call it a basic law of physics. We began the text with the observation that you cannot detect uniform motion. Michaelson and Morley thought they could, repeatedly tried to do so, and failed. This failure is known as the principle of relativity which Einstein used as the foundation of his theories of relativity. Throughout the text we have seen the impact of this simple idea. When combined with Maxwell’s theory of light, it implied that light traveled at the same speed relative to all observers. That implied moving clocks ran slow, moving lengths contracted, and the mass of a moving object increased with velocity. This led to the relationship E = mc² between mass and energy, and to the connections between electric and magnetic fields. The simple idea that you cannot measure uniform motion has an enormous impact on our understanding of the way matter behaves.
Now, with the particle-wave nature of matter, we are encountering an equally universal restriction on what we can measure, and that restriction has an equally important impact on our understanding of the behavior of matter. Our discussion of the uncertainty principle comes at the end of the text rather than at the beginning only because it has taken a while to develop the concepts we need to explain this restriction. With the principle of relativity we could rely on the student’s experience with uniform motion, clocks and meter sticks. For the uncertainty principle, we need some understanding of the behavior of particles and waves, and as we shall see, Fourier analysis plays an important role.
There are two forms of the uncertainty principle, one related to measurements of position and momentum, and the other related to measurements of time and energy. They are not separate laws, one can be derived from the other. The choice of which to use is a matter of convenience. Our discussion of the two slit experiment and the de Broglie relationship naturally leads to the position-momentum form of the law, while Fourier analysis naturally introduces the time-energy form.