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

Is color real? (Part 2)

Are tiles A and B the same color?

In my previous post I raised the question of whether color is a property of things in the world (such as paint, strawberries and fire-engines) or whether it is a property of the way that humans see things.

In everyday usage, we generally associate color with things in the world. We suppose that the fire-engine is red not merely when we look at it but also when we are not.

That has raised some scientific and philosophical issues. One issue is that sometimes people name colors differently because of vague boundaries. That seems to be the case with my disagreement with Jason. The color of the stool was on the boundary of red and orange. Since we learn about objects and their colors by looking at them and categorizing them, it is possible that people categorize colors slightly differently.

Another issue is that physical differences in our eyes can cause perceptual differences. This is the case with color blind people who can't differentiate between red and green.

A further issue is that background lighting influences the color we perceive. This is primarily the case with "the dress" phenomenon where some people report the dress as being blue-and-black and others that it is white-and-gold. It turns out that people see the actual dress as blue-and-black so the phenomenon only occurs when viewing an image of the dress. The colors people see generally depends on how much background is shown in the image and also the external lighting when viewing the image on a computer screen.

Then we get the issues of light reflection. If a paint can is closed (or it is open at night with the lights off), then paint would not appear red since color perception requires light to reflect off the surface of the paint onto our eyes. And, of course, a blind person never sees the paint as red.

Finally, there are the wavelengths of light. Visible light covers a small range of wavelengths in the electromagnetic spectrum and constitutes most of the light that the sun radiates. So color can be associated directly with the wavelength as well.

So there are a lot of things to consider. From a pragmatic point-of-view, we don't want to have to deal with all these issues (and others we haven't thought of) every time we refer to color in our everyday lives. We just want to say that the fire-engine is red regardless of the background lighting or the perceptual capabilities of different people.

The way we have achieved this with our everyday concept of color is by abstracting away all the complicating details. We simply associate color with the object [1]. If there are differences in perception due to lighting or a person's perceptual capabilities, then we use the word "appears" to make that distinction. Thus, the fire engine is red, but it doesn't appear red at night or to a blind person. The paint in the closed can is also red even though there is no light reflecting off the paint inside the can.

The benefit to abstracting in this way is that not only is the concept simple to learn and intuitive to use, it is also independent of any particular physical explanation for why color appears as it does. If future scientific discoveries are made that replace our best current theories of light, the reflective properties of surfaces, our eyes, or how our brain implements perception, then our everyday language around color would not need to change. We would just know more about color than we did before. [2]

So this is why color is a property of things and not a property of how we perceive things.

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[1] We learn concepts ostensively by paradigmatic examples. In this case, we learn colors by pointing at objects like fire-engines under normal lighting and comparing and contrasting them with other colored objects. Color is real (it is about objects in the world, not our perception of objects) and abstract (it need not carve nature at its joints).

[2] A further consideration is that other creatures perceive colors differently to humans as a consequence of their different brain and eye structures. So the scope and context of color concepts needs to be recognized.

Yes, they are the same color - they just appear different

Is color real?

I had a metaphysical discussion with my two-year old son this morning. I put a stool back in the room where it belonged (the chair kind of stool, not the other kind).

My son says, "Want orange stool."

So I retrieve the stool and say, "Jason, it's not an orange stool, it's a red stool."

He says, "Orange."

I go and get an obviously red drum stick and an obviously orange toy building. Showing him the drum stick, I say, "Is this red?"

"Yes."

Pointing to the toy building, I say, "Is this orange?".

"Yes."

Pointing to the stool, I say, "So Jason, see how the color of the stool is like the color of the red drum stick? So the stool is red."

"No, orange."

So my son and I are clearly having a serious metaphysical disagreement. Surely the stool has to be one color or the other. Can the stool be orange for him merely because he thinks it is orange?

Before you take the objective or subjective side of the dispute, here is a further puzzle about color. When we buy a can of paint, it has the name of a color on the label, such as "Red". However what is the color of the paint inside the can when the lid is on? (Note: there is no light inside the closed can). Also, when the can is opened under normal lighting, what is the color of the paint for a blind person?

For my solution, see Part 2.