Heisenberg's Uncertainty Principle

Diagram 1 - Hydrogen atom

In everyday life we are used to the idea of things like cars and trees having a precise velocity and position. For example, a car could have a velocity of 10km/h southwards and be currently leaving my driveway. And a tree is next to my driveway and not moving.

However it turns out that in our quantum universe the precision of the car's position is inversely related to the precision of the car's momentum (which is mass times velocity).^[1] Same for the tree's position and momentum. This imprecision is miniscule at human scales which is why we normally don't notice it. The inverse relationship between position and momentum is captured by Heisenberg's uncertainty principle.^[2]

The word "uncertainty" suggests that we can't accurately determine both values at the same time. However the principle is stronger than that. It means that the particle does not have an exact position and an exact momentum at the same time. This is somewhat analogous to how a bucket of water can't be completely solid and completely liquid at the same time.

To understand what this means physically, observe Diagram 2.

Diagram 2 - Wave superposition

The top four matter waves are sine waves from which a precise particle momentum can be derived from the wavelength of each wave.^[3] However amplitude for the particle position is concentrated in equal amounts along the entire wave. So the particle position for each wave is undefined (or, put differently, a particle is in a superposition of all possible positions when it has a precise momentum).

The bottom wave is formed by combining the top four waves in a superposition. In this case, the wave is more pronounced in the middle where the peaks and troughs are larger. While amplitude for the particle position is still spread out along the wave, it is no longer present in equal concentrations. It is more likely that a position measurement will locate a particle at the center of the wave (with the probability equal to the squared modulus of the amplitude).^[4] Note that since the bottom wave does not have a single wavelength, it does not have a precisely-defined particle momentum. A momentum measurement will return the momentum of one of the top four waves (resulting in decoherence and, consequently, indeterminacy in the particle's position).

Diagram 3 - Pulse wave

In the extreme case where all possible momenta are added, the result is a pulse wave as in Diagram 3. The particle has a precisely-defined position (since the amplitude of the horizontal lines is zero) but the momentum is undefined (i.e., the particle is in a superposition of all possible momenta).^[5]

In summary, the Heisenberg Uncertainty Principle states that a particle's position and momentum cannot both be sharply defined at the same time. This is because the range of momenta are derived from the wave's component wavelengths while the range of positions are derived from the concentrations of amplitude. A matter wave cannot be both a sine wave (precisely defining momentum) and a pulse wave (precisely defining position) at the same time.

For my first post in this series, see Visualizing Quantum Mechanics.

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Diagram 4 - Fourier transform

[1] Specifically, position and momentum are conjugate variables. This means that a matter wave can be represented in terms of position or equivalently, via a Fourier transform, in terms of momentum.

Another example of conjugate variables are frequency and time. Consider Diagram 4, where a sound signal over time has been recorded in the red box. A Fourier transform decomposes the signal into three uniform waves and produces a new signal in the blue box. An inverse Fourier transform can be used to recreate the original time-based signal.

Our ears perform a similar function to this when they receive sound waves as air vibrations and transform them into separate frequencies that are sent to the brain.

[2] The equation for Heisenberg's Uncertainty Principle is ΔxΔp ≥ h/4π where Δx and Δp are the standard deviations of position and momentum, and h is Planck's constant (10^-34). The indeterminacy in the position of a thrown baseball is 10^-30 millimeters. This indeterminacy becomes significant at tiny scales such as shown in Diagram 1 where the position of the electron in a Hydrogen atom is represented as a cloud.

[3] The de Broglie wavelength: λ = h/p where p is momentum.

[4] This is the Born rule. Note that the probability must be normalized so that the sum of the squared moduli of the probability amplitudes of all the possible positions is equal to one. For a sense of how the bottom wave in Diagram 2 would be represented in three dimensions, imagine the wave as a corkscrew winding along and around the x-axis with a larger bulge where the higher amplitude is concentrated.

[5] This is nicely demonstrated by Walter Lewin in this single-slit experiment video. As the slit becomes narrower, the position of the photons in the slit become more precisely defined which results in a wider spread of photons on the back screen (since direction is a component of momentum). See also Diagram 5 below. The reason why this wider spread of photons occurs is because amplitude flows from all positions in the slit to all positions on the back screen but a wider angle from the narrow slit is needed for destructive interference to occur.

Diagram 5 - Single-slit experiment