The Limits of Precision: Navigating Heisenberg’s Uncertainty Principle

Heisenberg’s Uncertainty Principle

     Let us say we are conducting an experiment where we are sending electrons through a narrow slit. In a certain distance behind the slit, there is a detection plate, which detects the position of individual electrons that strike it. We already know from the previous chapters that we cannot predict where any individual electron ends up on the plate (because of superposition). We can, however, know the probability of an electron ending up in a certain place on the plate if we know its wave function.

Scheme of the experiment – electrons are sent through a narrow slit and strike the detection plate. The graph shows that the vast majority of electrons strike the area directly behind the plate. The grey color shows the area where the majority of electrons are.

 

    If we make the slit smaller, we probably intuitively expect the electrons to fall into a narrower section on the plate. Let us say we start with a relatively wide opening which we taper gradually. At first, our prediction is correct and the electrons indeed start falling into an increasingly narrower section. At some point, however, the opposite begins to happen. If one continues to make the slit smaller to the point where it is considerably narrow, the electrons start spreading again.

    

When the slit is narrowed considerably, electrons start to spread on the plate. The majority of electrons now do not end up directly behind the slit.

 

    This phenomenon is a consequence of the so-called Heisenberg uncertainty principle, which was introduced by Werner Heisenberg in 1927. The uncertainty principle states that there are pairs of physical properties whose precise values cannot be known simultaneously. The more precisely we know one property, the more uncertainty there is about the other property. The most famous pair of such properties is momentum and position. The uncertainty in the momentum of a given particle multiplied by the uncertainty in the position of this particle is always equal or greater than the value of the reduced Planck constant divided by two:

𝚫𝐱 ⋅ 𝚫𝐩 ≥ ħ/𝟐

    The more accurately one knows the position of a particle, the less information one has about its momentum. Let us go back to the electrons going through a slit. If we make the slit narrower, the uncertainty about the position of the electrons is decreased. Consequently, the uncertainty about their momentum has to be increased. The electrons now have a greater probability of changing their direction (i.e. are deflected sideways) or velocity, leading to them being more spread on the plate.

    The Heisenberg uncertainty principle is a mere consequence of the wave function. Let us consider, for example, that we want to measure the momentum of a certain particle as accurately as possible. De Broglie’s equation (λ =h/p) shows that the momentum of a particle depends on the wavelength of its wave function (p =h/ λ). Therefore, if we want to ascertain the wavelength, the wave function cannot be too localized, since the wavelength of a localized wave is not precisely determined. On the other hand, if we want to measure the position of a particle, we need a wave that is as localized as possible. Of course, a wave cannot be both localized and spread simultaneously, which means that when measuring the position and the momentum of a particle at the same time, one has to find a compromise in the form of a wave function that is partially localized and partially spread and as such provides relatively precise values for both position and momentum. Such a wave function is called a wave packet.

An ideal wave function to determine the momentum of an object (spread). The uncertainty regarding the position is huge. Its wavelength is precisely known.

 An ideal wave function to determine the position of an object (localized). The uncertainty regarding the momentum is huge. Its wavelength is completely unknown.

An example of a wave packet – the wave function is partially localized and partially spread.

 

    The uncertainty principle is often mistakenly interchanged with the so-called observer effect, which is a phenomenon that occurs every time a physical system is observed. The observer effect states that any time a system is observed, its state inevitably changes. For example, when ascertaining the position of an object using our vision, photons have to bounce off the object into our eyes, so its position is not the same as it had been before the observation occurred. This phenomenon, however, has nothing to do with the uncertainty principle, since the uncertainty in the position and the momentum of a quantum object exists all the time, regardless of the presence of an observer. We can basically say that even the object itself does not “know” its own position and momentum simultaneously. Therefore, explaining the uncertainty principle using the observer effect is wrong.