Modeling quantum interference

Diagram 1: Mach-Zehnder interferometer

In this post, I'm going to model a device that exhibits quantum behavior in a simple but striking way.

The device pictured at the left is called a Mach-Zehnder interferometer. The beam splitter splits a beam of light into two paths. 50% of the beam is reflected towards mirror 1 and 50% of the beam is transmitted towards mirror 2. When the beams reach the second beam splitter, each beam is split again and is reflected or transmitted towards the detectors.

Intuitively, it would seem that half the light should end up at detector 1 and half at detector 2.^[1] However, assuming the two paths are the same length, all the light actually ends up at detector 1 at the right and none at the top detector.

This result is due to quantum interference at the second beam splitter where light heading towards the top detector destructively interferes and light heading towards the right detector constructively interferes. In true quantum style, this result always occurs even if only a single photon of light is emitted towards the first beam splitter.

In quantum mechanical terms, the photon is in a superposition of travelling along both paths simultaneously. At the second beam splitter, each path forms a further superposition (again with one path reflecting and one transmitting - see the four arrows heading towards the detectors in the diagram). The two paths heading towards detector 2 destructively interfere (i.e., they are 180° out of phase) and thus cancel each other out. Whereas the two paths heading toward detector 1 constructively interfere and so the photon is always detected there.

So how does the device actually work? The mathematics is actually fairly straightforward. The basic strategy is to model each path that the photon can take and combine identical paths at the end. Each path segment has a complex value associated with it called an amplitude which can be visualized as an arrow that can rotate around a center point (like a clock hand).^[2]

The initial (blue) path amplitude is 1 (see Diagram 2 below which specifies the calculated amplitudes for each path segment). The basic rule at the beam splitter is that the path splits into two paths and each path takes the amplitude of the source path value and multiplies it by 1/√2 (this is the normalization condition - the squares of the amplitudes in a superposition of paths must always sum to 1, i.e., 1/2 + 1/2 = 1).^[3] Also, the path of the reflected beam additionally multiplies the amplitude by -1 which represents a phase change of 180°. So the upper (green) path has an amplitude of -1/√2 (-0.707) and the lower (red) path has an amplitude of 1/√2 (0.707).

(Note: If the photon passes through the rear of the beam splitter, the result is the same except that the phase change rule does not apply.^[4] This is the case for the upper beam path when it reaches the second beam splitter. The front of each beam splitter is indicated by the dot.)

At each mirror, the amplitude for each path is multiplied by -1 (i.e., a phase change of 180°). So the upper path now has a value of 1/√2. The lower path now has a value of -1/√2. At the second beam splitter, the upper (reflection) path itself splits into reflection and transmission paths toward the two detectors. The upper beam reflection path value is 0.5 (1/√2 * 1/√2) and the upper beam transmission path value is also 0.5 (1/√2 * 1/√2). The lower beam reflection path value is 0.5 (-1/√2 * 1/√2 * -1) and the lower beam transmission path value is -0.5 (-1/√2 * 1/√2).

Diagram 2: Path amplitudes

This is where the quantum magic happens. The upper beam reflection path and the lower beam transmission path coincide. They are both directed towards detector 2. So the paths merge and the amplitudes are added to give a value of 0 (0.5 + -0.5) which is destructive interference. Similarly, the upper beam transmission path and the lower beam reflection path also coincide. They are both directed towards detector 1. So the paths merge and the amplitudes are added to give a value of 1 (0.5 + 0.5) which is constructive interference.

The probability of finding the photon at a particular detector is given by the amplitude squared, which is 100% for detector 1. Thus the photon always ends up at detector 1.

Note that this result depends on the physical configuration of the interferometer. In this case, the two paths between the beam splitters are the same length. However if the length of one of the paths is changed, the results also can change such that the photon is instead always found at detector 2 (i.e., change a path phase by 180° by multiplying by -1 and recalculate the subsequent path values), or found at either detector with equal probability (i.e., change a path phase by 90° by multiplying by the imaginary number i and recalculate), or any other probabilistic combination.

For further interferometer fun, see Part 2.

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[1] If the second beam splitter were removed, the light would be distributed between both detectors. In the case of one emitted photon, the photon would be observed at one detector or the other with 50% probability. No interference between the photon paths would occur since the paths are different (one is directed towards detector 1 and one is directed towards detector 2 when they cross).

[2] The amplitude actually continually changes as the photon travels (i.e., the arrow rotates). To simplify the example, the path segments are of lengths that are multiples of the wavelength. So a photon that leaves the beam splitter with a particular phase angle will have the same phase angle when it arrives at the mirror. Also, the top path and lower path are the same length.

[3] This is in accordance with the Born rule. The probability that the photon will be observed on a particular path is given by the square of the amplitude.

[4] There is a phase change for a reflection at a surface with a higher refractive index which is true at the front of the beam splitter (the glass refracts more than the air the photon is travelling in) but not at the rear (where the photon is already travelling in the glass before it reflects).

Visualizing the Schrodinger equation

The Schrodinger equation describes how a physical system changes over time. But what does it mean intuitively?^[1]

Imagine a particle moving freely through space. There are no forces acting on the particle so it travels in a straight line along the x-axis.

In Classical Mechanics, if the current state of the particle is known (such as its position and momentum) then its future state can be predicted according to classical laws.

In Quantum Mechanics, the state of the particle is represented by a wave function^[2] that has a complex value and is denoted by the Greek letter Ψ (psi). The state of the particle can be prepared so that its wave function is initially known and it will then evolve in time according to the Schrodinger equation. Further mathematical operations can be performed on the wave function to determine the position, momentum or energy of the particle.

The time evolution can be elegantly expressed as Ψ(t) = U(t)Ψ(t=0) where U(t) is a unitary operator that propagates the wave function from its initial configuration at time t=0 to its final configuration at time t. U(t) = e^-iEt/ħ which is an exponential formula that represents a rotation (or phase change) on the complex plane of Et/ħ radians, where E is the total energy of the system and ħ is the reduced Planck constant. The wave function (at time t=0) can be visualized as a clock hand that rotates to a new position when the operator U(t) is applied to it. The greater the energy, the more rotations per second.

So we know U(t) and how to calculate the future wave function Ψ(t). All we need is the wave function at time t=0 to plug into the equation. The simplest wave function is a plane wave that curls in a uniform spiral around the x-axis (see Figure 1 below).

Figure 1 - Complex plane wave

A plane wave has the general formula Ae^ipx/ħ where A is the wave amplitude, p is the momentum and x is the position on the x-axis. Note that, like the time evolution operator, it also has an exponential representing rotation on the complex plane. However, in this case, the rotation is across space rather than over time. The greater the momentum, the more rotations per meter.

So our initial wave function is Ψ(x,t=0) = Ae^ipx/ħ. Therefore our wave function at time t is Ψ(x,t) = e^-iEt/ħAe^ipx/ħ, which can be expressed more simply as:

Ψ(x,t) = Ae^{i(px - Et)/ħ} 

where the energy is proportional to the square of the momentum (E=p²/2m) and position and momentum are related via the canonical commutation relation xp - px = iħ.^[3]

A way to visualize this equation is to imagine the entire wave in Figure 1 to be dynamically rotating as time progresses. It will appear to be travelling along the x-axis in a periodic manner. The greater the momentum (and therefore energy), the tighter the spiral and the faster it will be spinning.

Let's suppose that we've prepared our particle to have a precise momentum (for example, 40 kgm/s where our particle weighs 5kg and has an energy of 160 joules, since E=p²/2m). So our wave function Ψ is Ae^{i(40x - 160t)/ħ}. We can now use our operators to measure those observable quantities in our wave function.

Let's start with momentum. The operator for measuring the particle's momentum is -iħ ∂/∂x. So -iħ ∂Ψ/∂x = -iħ(i40/ħ)Ψ = 40Ψ. Since the result is a constant times Ψ, Ψ is an eigenfunction of the momentum operator with eigenvalue 40. So if we make a measurement, we will measure the momentum of the particle to be 40 kgm/s with certainty. In this case the shape of the momentum function is the same as Ψ, but its amplitude is everywhere scaled by 40.

Let's try energy. The energy operator is iħ ∂/∂t. So iħ ∂Ψ/∂t = iħ(-i160/ħ)Ψ = 160Ψ. So Ψ is an eigenfunction of the energy operator with eigenvalue 160. If we make a measurement, we will measure the energy of the particle to be 160 joules with certainty. As with momentum, the shape of the energy function is the same as Ψ, but its amplitude is scaled by 160.

Now we'll try the position. The position operator is simply x. But xΨ is not a constant times Ψ, so Ψ is not an eigenfunction of position. This means there is uncertainty about the position of the particle. The position function scales the amplitude of Ψ at each x-position by the ordinal value of that x-position (i.e., the spirals increase in amplitude along the x-axis like a cone). (Note that the eigenfunctions of the position operator are actually Dirac delta functions which spike at their respective x-positions and have zero amplitude everywhere else.)

We can actually predict these results by looking at the plane wave in Figure 1. It has a single wavelength which translates to a single momentum and energy (per the de Broglie relations). But it has the same amplitude everywhere so the particle's position is spread through spacetime. A wave function that provides a more localized position is a wave packet that combines a range of plane waves of different frequencies in superposition as shown in Figure 2.^[4] The wave packet also curls in a spiral as it propagates along the x-axis over time, but it has a localized distribution of non-zero amplitude as illustrated in Figure 2 below.^[5]

Figure 2 - Wave packet

This diagram represents a wave function at a single instant in time. Each white ball represents the complex amplitude of the wave at a particular x-position. This amplitude can be used to calculate the probability that a measurement will find the particle at that x-position.^[6] Note that x-positions beyond the two ends of the wave packet have zero amplitude which means that the particle cannot be located at those x-positions.

Suppose that the particle were prepared in a superposition of Ψ = Ψ₁ + Ψ₂ = A₁e^{i(p₁x - E₁t)/ħ} + A₂e^{i(p₂x - E₂t)/ħ}.^[7] For example, the amplitudes are A₁=6 and A₂=8 and the momenta are p₁=5 kgm/s and p₂=3 kgm/s.

If we apply the (linear) momentum operator, we get -iħ ∂/∂x(Ψ₁ + Ψ₂) = p₁Ψ₁ + p₂Ψ₂ (i.e., 5Ψ₁ + 3Ψ₂). So Ψ₁ and Ψ₂ are both eigenfunctions of the momentum operator with different momentum eigenvalues.

Now suppose a momentum of 3 kgm/s is measured in an actual experiment locating the experimenter with Ψ₂. If measurement is a linear process, the particle's momentum would also be measured to be 5 kgm/s locating the experimenter with Ψ₁.

The fact that the experimenter reports only one measurement outcome (with a probability ratio of A₁²:A₂² which is 36:64 in our example) is what gives rise to the measurement problem. The Copenhagen Interpretation postulates that the wave function collapses to Ψ₂ (Ψ = Ψ₂) and Ψ₁ disappears. Whereas the Many Worlds Interpretation assumes that the wave function Ψ continues to evolve unitarily with the experimenter and measuring apparatus now entangled with the particle in superposition.

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[1] This post builds on the concept of exponential growth in the complex number plane that was explored in Visualizing Euler's Identity.

[2] Familiar examples of wave behavior are vibrating guitar strings and ocean waves. Quantum Mechanics applies this idea to all matter via the de Broglie hypothesis and so the Schrodinger equation is a wave equation that describes how matter waves evolve. For an excellent tutorial on wave equations, see here.

Note that it is important to distinguish between the quantum state and the wave function. A wave function is a representation of a quantum state in a particular basis, such as position or momentum. This post presents the wave function in the position basis (as plane wave states of definite momentum). It can be alternatively represented in the momentum basis via a Fourier transform.

[3] Given the plane wave solution Ψ = Ae^{i(px - Et)/ħ}, the Schrodinger equation can be derived. The time derivative ∂Ψ/∂t is how the wave function changes over time and is (-iE/ħ)Ψ. The first spatial derivative ∂Ψ/∂x is how the wave function slopes in space and is (ip/ħ)Ψ. The second spatial derivative ∂²Ψ/∂x² is how the wave function curves in space and is (i²p²/ħ²)Ψ which reduces to (-p²/ħ²)Ψ.

Multiplying the time derivative by iħ gives iħ ∂Ψ/∂t = iħ(-iE/ħ)Ψ = EΨ. Multiplying the second spatial derivative by -ħ²/2m gives (-ħ²/2m)∂²Ψ/∂x² = (-ħ²/2m)(-p²/ħ²)Ψ = (p²/2m)Ψ = EΨ (E=p²/2m relates kinetic energy to momentum).

Therefore -ħ²/2m ∂²Ψ/∂x² = iħ ∂Ψ/∂t which is the time-dependent Schrodinger equation for a free particle in one dimension. The equation for any non-relativistic particle is described in Figure 3 below. Note that ∇² represents the second derivative over all space (x,y,z) and that V represents the potential energy which, for free particles, is zero.

Figure 3 - Time-dependent Schrodinger equation for a single non-relativistic particle

The Schrodinger equation expresses the principle of the conservation of energy consistent with the de Broglie relations. That is, the kinetic energy of the particle (which is proportional to the curvature of Ψ over space) plus the potential energy of the particle equals the total energy (which is proportional to the slope of Ψ over time).

The time-dependent Schrodinger equation can be more generally expressed as iħ ∂Ψ/∂t = ĤΨ where Ĥ is the Hamiltonian operator (representing the total energy of the system) and differs with the situation or number of particles. In our free particle scenario where the potential energy is zero, Ĥ = -ħ²/2m ∂²/∂x².

The simpler time-independent Schrodinger equation applies to stationary states and is ĤΨ = EΨ where E is the total energy of the system. This is an eigenvalue equation which means that the Hamiltonian operates on the function Ψ and produces a definite (and real) energy value E multiplied by the same function Ψ. If Ψ describes the physical system and satisfies the eigenvalue equation (meaning it is an eigenfunction), then that energy eigenvalue would be measured with 100% certainty. In general, Ψ will not be an eigenfunction of the Hamiltonian but, instead, will be a linear superposition of energy eigenfunctions (with the probability of measuring a particular energy eigenvalue being the squared magnitude of the amplitude of that eigenfunction, per the Born rule).

[4] Combining a finite number of plane waves also fails to avoid the problem of the particle being delocalized since the large wave packets will still be periodic through spacetime (with other smaller periodic wave packets in between). It is only in the limit that there is a single wave packet as other wave packets would, in effect, be infinitely far away. That is, an integral over a continuous range of wave numbers (or momenta, since p=kħ) produces a single localized wave packet.

[5] Figure 2 (enlarged below) is a snapshot of a localized particle's wave function at an instant in time. Per Euler's formula, e^{i(px - Et)/ħ} = cos((px - Et)/ħ) + i.sin((px - Et)/ħ). So the complex spiral is the sum of the real cosine wave (at the back) and the imaginary sine wave (at the bottom) propagating along the x-axis. Each white ball represents the amplitude of the complex wave at that particular x-position (imagine the clock hand pointing laterally from the x-position on the x-axis to the white ball). Instead of visualizing a ball moving with the wave packet along the x-axis as time progresses, imagine that it remains at the same x-position, but simply spins around the x-axis in the complex plane, shrinking or expanding in magnitude as time progresses (i.e., as the wave packet propagates through that x-position).

Now imagine that the white ball is actually a linear combination of colored balls at that x-position, one for each plane wave in the superposition (and each with a different magnitude and phase). Each colored ball simply spins around the x-axis with a fixed magnitude but the colored balls taken together constructively and destructively interfere to produce the white ball that is seen in the image as the wave packet propagates through that x-position. That is, the entire propagating wave packet can be explained as a combination of fixed length clock hands spinning at different rates.

Figure 2 (enlarged) - Wave packet

[6] The probability that a measurement will find the particle at a particular x-position is calculated by squaring the magnitude of the wave function's amplitude at that particular x-position and time per the Born Rule. The amplitude is a complex number that, when multiplied by its complex conjugate, produces a real number that is the square of the magnitude. The squared magnitude is also the intensity of the wave function at that position and time.

[7] Ψ is a combination of different momentum basis states e^{i(p_nx - E_nt)/ħ}, each with its own coefficient (amplitude) A_n where n is the index for the basis state.

Bell's Theorem

Bell's Theorem shows that no physical theory that depends on precisely existing values for unmeasured quantities (termed local hidden variables) can reproduce all of the predictions of quantum mechanics. However the Many Worlds Interpretation does not depend on hidden variables so how does it produce the correct predictions?

Consider a series of experiments with each experiment involving an entangled photon pair named A and B respectively. In each experiment, the photons will each be measured by polarized filters which can be set to any angle from 0° to 360°. When a measurement is taken, the photon will either align with the filter and therefore be transmitted or it will not align and therefore be absorbed or reflected.^[1] The entangled photons have been prepared so that if they are measured by polarizers set to the same angle then they will either both be transmitted or both absorbed, with a 50% chance of observing either outcome.

The amplitude for configurations where the photons are correlated (i.e., have the same alignment) is the cosine of the angle between the two polarizers. Conversely, the amplitude for configurations where the photons are anti-correlated is the sine of the angle. In our experiments, the polarizers are set to either 0°, 120° or 240°.

For the first experiment, the two polarizers are set to 0°. The configurations (along with amplitudes) that are in superposition prior to measurement are as follows:

0°-1. Photon A is aligned and photon B is aligned (cos 0° = 1)
0°-2. Photon A is aligned and photon B is non-aligned (sin 0° = 0)
0°-3. Photon A is non-aligned and photon B is non-aligned (cos 0° = 1)
0°-4. Photon A is non-aligned and photon B is aligned (sin 0° = 0)

When the photons are measured, different versions of the observer become entangled with each configuration that contains amplitude. The probability that an observer will find themselves entangled with a particular configuration is determined by the squared modulus of the amplitude.^[2] So this is 50% each for configurations 1 and 3 and 0% for the other two configurations. That is, there is a 100% probability that the photon pair will have the same alignment (i.e., be correlated).

The same configurations apply if both polarizers are set to 120° or 240° respectively since the angle between the polarizers is still 0°. However what happens if the polarizer for photon A is set to 0° and the polarizer for photon B is set to 120°? In this case, the configurations in superposition are:

120°-1. Photon A is aligned and photon B is aligned (cos 120° = -0.5)
120°-2. Photon A is aligned and photon B is non-aligned (sin 120° = 0.866)
120°-3. Photon A is non-aligned and photon B is non-aligned (cos 120° = 0.5)
120°-4. Photon A is non-aligned and photon B is aligned (sin 120° = 0.866)

The probabilities for configurations 1 and 3 are 12.5% each and 37.5% each for configurations 2 and 4. That is, there is a 25% probability that the photon pair will have the same alignment and a 75% probability that it will not. The same configurations apply if the polarizer for photon A is set to 120° and the polarizer for photon B is set to 240° since the angle between the polarizers is still 120°. Similarly if the polarizers are set to 240° and 0° respectively.

The same probabilities also occur if the polarization angle difference is set to 240°. However the configurations are slightly different, as follows.

240°-1. Photon A is aligned and photon B is aligned (cos 240° = -0.5)
240°-2. Photon A is aligned and photon B is non-aligned (sin 240° = -0.866)
240°-3. Photon A is non-aligned and photon B is non-aligned (cos 240° = 0.5)
240°-4. Photon A is non-aligned and photon B is aligned (sin 240° = -0.866)

Note that if the polarizers are set randomly, then the average probability that the measurements will correlate is 50%. That is, (100% + 25% + 25%) / 3 = 50%. Similarly, for measurement anti-correlation, (0% + 75% + 75%) / 3 = 50%.

Now suppose that the entangled photons are light years apart and the polarizer angles are set randomly just prior to measurement. Observers can only be entangled with one of the configurations listed above, so measurements will continue to conform to the specified probabilities.

So Many Worlds produces the correct predictions. However the above scenario is impossible to replicate by any single-world theory that depends on local hidden variables. Such theories entail that a photon has a definite polarization for each of the three angles 0°, 120° and 240° prior to measurement. For this to be true, either the photon has the same alignment value for all three angles (two options - AAA and NNN) or it has two alignment values the same and one different (six options - AAN, ANA, NAA, NNA, NAN, ANN).

Now if we could measure any pair of polarization values, they would either correlate 100% of the time for the first two options or 33% of the time for the last six options. For example, with AAN, the pairings are AA, AN and AN, with only AA correlating. So overall the measurements would correlate at least 33% of the time. This is called a Bell inequality.

It so happens that this inequality can be tested by measuring a pair of photon values via entanglement. To do this, the first polarizer angle is set to 0° and the second polarizer angle is set to 120°. Since photon A and B have identical attributes (i.e., they always correlate when measured at the same angle) then the value measured for photon B at 120° is the value that would have been the value measured for photon A at 120° if it had been measured. As was shown above, the pair of values for photon A at 0° and 120° should correlate at least 33% of the time and so the actual measured values for photon A and photon B should also correlate at least 33% of the time. However, as shown earlier with the 120° configurations, Quantum Mechanics predicts that they will correlate only 25% of the time (cos² 120° = 0.25). So Quantum Mechanics violates the Bell inequality. Therefore photons cannot have precisely-defined values for angles that have not been measured.

One possible loophole remains and that is to ensure that the entangled photons do not somehow communicate with each other. This loophole can be closed by ensuring that the photons are space-like separated when the polarizer angles are set and the measurements performed (i.e., at a distance that would require superluminal communication between the photons). The results are that experiments support the predictions of quantum mechanics and definitively rule out local hidden variable theories.

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[1] This is how polarized sunglasses work. Sunlight normally radiates in all directions. However sunlight that reflects off flat horizontal surfaces such as lakes and roads will be polarized horizontally resulting in a strong glare that vertically polarized sunglasses can block.

[2] This is the Born rule. Note: In order to keep the tables simple, the configuration amplitudes have not been normalized (i.e., the modulus squared probabilities do not add to 1). To normalize, multiply each amplitude by 1/√2.

Three sides to every story

Have you ever imagined yourself as the hero in a story where, against all odds, you take on the villains to save the world? Feels good doesn't it?

Politics provides an opportunity to take part in such a story. The villains consist of those that conspire to thwart your team's noble cause, either by their actions or their indifference.

For conservatives, the battle is between civilization and barbarism. Conservative themes that fit this narrative include terrorism, socialism and moral issues like abortion.

For progressives, the struggle is for oppressed or marginalized groups and include concerns such as inequality, racism and corporate power.

For libertarians, the conflict is coercion against liberty, particularly with respect to the role of government in the lives of individuals and communities, such as the extent of trade regulations, taxes and legislated personal morality.

Any political issue can be described in terms of these axes.[1] For example, illegal immigrants can be regarded as experiencing oppression, importing barbarism, or seeking freedom depending on the chosen framing narrative. In a particular election cycle, the party that tells the most persuasive story around the political issues of the day will win the election. It is not the facts and policies themselves that matter for winning the vote but, instead, how those facts and policies are framed.

In the upcoming US election, Donald Trump's slogan is "Make America Great Again". This fits particularly well with the civilization/barbarism narrative and taps into people's sense that things aren't so great right now. It also makes it easy for anyone to create their own vision for what America's greatness means. It's a difficult slogan for Democrats to attack. So they instead need to challenge Trump's credibility for delivering on it.

Hillary Clinton's slogan is "Stronger Together". This strikes a solidarity chord with progressives. But it doesn't speak to people's aspirations or struggles. So Clinton's challenge is to find a message that emotionally resonates with people.[2]

Libertarian candidate Gary Johnson's slogan is "Live Free". This fits the libertarian narrative. But, as with Clinton's slogan, it will miss the mark with most voters. Libertarians need to inspire independents who are not enamored by either of the major party candidates. So how would freedom lead to a more civilized, safe and just society or improve their day-to-day lives?

Terrorist threats and economic hardship feed into Trump's story that there are systemic problems that only he, the uncompromising Washington outsider, can solve. Unless his opponents create a persuasive alternative story, Trump wins in November.

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[1] See Arnold Kling's Tribal Politics in the 21st Century.

[2] See Scott Adam's Battle of the Campaign Slogans.

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.