28.7 Probability: the heisenberg uncertainty principle  (Page 2/11)

Heisenberg uncertainty

How does knowing which slit the electron passed through change the pattern? The answer is fundamentally important— measurement affects the system being observed . Information can be lost, and in some cases it is impossible to measure two physical quantities simultaneously to exact precision. For example, you can measure the position of a moving electron by scattering light or other electrons from it. Those probes have momentum themselves, and by scattering from the electron, they change its momentum in a manner that loses information . There is a limit to absolute knowledge, even in principle.

It was Werner Heisenberg who first stated this limit to knowledge in 1929 as a result of his work on quantum mechanics and the wave characteristics of all particles. (See [link] ). Specifically, consider simultaneously measuring the position and momentum of an electron (it could be any particle). There is an uncertainty in position     $\mathrm{\Delta }x$ that is approximately equal to the wavelength of the particle. That is,

As discussed above, a wave is not located at one point in space. If the electron’s position is measured repeatedly, a spread in locations will be observed, implying an uncertainty in position $\mathrm{\Delta }x$ . To detect the position of the particle, we must interact with it, such as having it collide with a detector. In the collision, the particle will lose momentum. This change in momentum could be anywhere from close to zero to the total momentum of the particle, $p=h/\lambda$ . It is not possible to tell how much momentum will be transferred to a detector, and so there is an uncertainty in momentum     $\mathrm{\Delta }p$ , too. In fact, the uncertainty in momentum may be as large as the momentum itself, which in equation form means that

The uncertainty in position can be reduced by using a shorter-wavelength electron, since $\mathrm{\Delta }x\approx \lambda$ . But shortening the wavelength increases the uncertainty in momentum, since $\mathrm{\Delta }p\approx h/\lambda$ . Conversely, the uncertainty in momentum can be reduced by using a longer-wavelength electron, but this increases the uncertainty in position. Mathematically, you can express this trade-off by multiplying the uncertainties. The wavelength cancels, leaving

So if one uncertainty is reduced, the other must increase so that their product is $\approx h$ .

With the use of advanced mathematics, Heisenberg showed that the best that can be done in a simultaneous measurement of position and momentum is