Modeling quantum interference (Part 2)

Diagram 1: Mach-Zehnder interferometer

In my previous post I described how to model a Mach-Zehnder interferometer by taking into account all paths that a photon can take and calculating the amplitudes. In this post, I will elaborate on the mathematical model further to create a visual sense of what is going on.

The interferometer and the photon travelling within it can be considered together as a quantum system. At the beginning of the experiment the system has a single quantum state that contains all the information about the system. This quantum state 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 or compass arrow).^[1]

In our model, the amplitude for the initial quantum state is 1. As the photon travels toward the first beam splitter, the quantum state continually changes and this change is reflected in the amplitude. Using the clock analogy, the clock hand is continually rotating as the photon travels. When the photon arrives at the beam splitter, the quantum state is transformed into two distinct quantum states which are in superposition - one representing a reflected photon and one representing a transmitted photon. Each quantum state has its own amplitude as determined by the photon interaction with the beam splitter. There are now three distinct quantum states. One state represents a photon travelling on the upper (green) path, one state represents a photon travelling on the lower (red) path and the global state that now represents those two component states in superposition.

To help conceptualize this, we can think of each quantum state as representing a real system. So we can refer to the photon on the green path, the photon on the red path, and the global photon that is in superposition, with each referent indexed to a distinct quantum system.^[2]

Mathematically, the complex amplitude that represents a quantum state is a vector and it allows us to model the superposition of the two quantum states. Consider an arrow pointing north-east. This can be thought of as a vector that is the combination of two basis vectors - one pointing north and one pointing east. Suppose the arrow is 1 unit long. The arrow therefore has a length of 1/√2 in the north direction and a length of 1/√2 in the east direction (per the Pythagorean theorem). So we can represent this as a linear equation using bra-ket notation:

  |north-east> = 1/√2|north> + 1/√2|east>

In the same notation, when the photon passes through the first beam splitter the new quantum state is:

  |blue_aftersplitter> = -1/√2|green> + 1/√2|red>

This captures all the information about the system at this point. |blue_aftersplitter> is the main quantum state that is in superposition with an amplitude of 1 (implied). The state for the green (reflected) path has an amplitude of -1/√2 and the state for the red (transmission) path has an amplitude of 1/√2. See diagram 2 below for the path amplitudes.

Diagram 2: Path amplitudes

Note that the squares of the two component amplitudes sum to 1. If a measurement were performed at this point (i.e., the detectors were placed on those two path segments), there would be an equal probability of finding the photon in either state.^[3]







Evolving from the initial (blue) quantum state to the superposition state after the beam splitter requires a transformation operation which is represented by a matrix. The matrix required for the beam splitter is:^[4]

  bs = [1/√2  1/√2] = 1/√2[1  1]
       [1/√2 -1/√2]       [1 -1]

The beam splitter has two input ports - one at the front (indicated by the dot) and one at the rear. The matrix columns describes the rear port and front port behavior respectively. The matrix rows describe the transformations on the transmitted beam and reflected beam respectively. (For example, the bottom-right cell describes the transformation on a photon that enters the beam splitter through the front port and is reflected.)

Since the photon initially passes through the front port, the initial (blue) state is described by the following vector which specifies an amplitude of 0 for the photon passing through the rear port and an amplitude of 1 for the photon passing through the front port:^[5]

  |blue> = [0]

           [1]

We can now see what happens when the initial (blue) state is transformed by the beam splitter matrix (i.e., when the vector is multiplied by the matrix).

  bs|blue> = 1/√2[1  1][0] = 1/√2[0*1 +  1*1] = 1/√2[ 1] = [ 1/√2]

                 [1 -1][1]       [0*1 + 1*-1]       [-1]   [-1/√2]

The top cell of the resulting vector describes the transmitted (red) state amplitude and the bottom cell describes the reflected (green) state amplitude. That is, the amplitude for the global (blue) state is now distributed between two component states (red and green). This can be equivalently expressed as:

  bs|blue> = 1/√2(-|green> + |red>)

           = -1/√2|green> + 1/√2|red>

To explain the above equation, we know the front port behavior is described by the second column of the bs matrix. Then the bottom cell applies to the reflected beam state (which, in this case, is green) so it is multiplied by -1. The top cell applies to the transmitted beam state (which, in this case, is red) so it remains unchanged. Finally, the 1/√2 coefficient applies to both states so they are both multiplied by it.

The transformation matrix required for the two mirrors is:

  mi = [-1  0]

       [0  -1]

which inverts the phase of both states (i.e., multiplies each state by -1). The second beam splitter has the same matrix as the first. Note that, in this case, a beam will pass through the rear of the beam splitter as well as a beam through the front.

The entire evolution from the initial quantum state is:

  bs mi bs|blue>
    = bs mi (-|green> + |red>)/√2
    = bs (-(-|green) + -(|red>))/√2
    = bs (|green - |red>)/√2
    = ((|detector1> + |detector2>) - (-|detector1> + |detector2>))/2
    = (|detector1> + |detector2> + |detector1> - |detector2>)/2
    = (2|detector1> + 0|detector2>)/2
    = |detector1>

  P(|detector1>) = |1|² = 1 = 100%

Thus the photon always ends up at detector 1. To explain the mathematics, the photon in the initial (blue) state passes through the front of the beam splitter so its reflected beam (green) is inverted. Both beams are then inverted by the mirrors. Finally, the green state itself becomes a superposition of two states representing the photon heading towards each detector. Similarly for the red state. But note that the photon in the red path passes through the front of the beam splitter, so its reflected beam to detector 1 is inverted. The detector 2 states destructively interfere (since they are indistinguishable) while the detector 1 states constructively interfere (since they are also indistinguishable) which finally results in a single quantum state at detector 1 with an amplitude of 1.

Note that it is also possible to insert a phase shifter into one of the paths (emulating a sample or change in path length) which will change the probability of the photon arriving at each detector. The transformation matrix for a phase shifter in the lower (red) path is:

  ph(φ) = [e^iφ 0]

          [0   1]

where φ is the phase angle. A phase shift of 180° means the photon will always arrive at detector 2 while a phase shift of 90° means the photon will be found at either detector with equal probability. To illustrate this, adding a phase shift of 90° gives:

  ph(90°) = [e^i*pi/2 0] = [i 0]

            [0      1]   [0 1]

The state evolution is now:

  bs mi ph(90°) bs|blue>
    = bs mi ph(90°) (-|green> + |red>)/√2
    = bs mi (-|green> + i|red>)/√2
    = bs (-(-|green) + -(i|red>))/√2
    = bs (|green - i|red>)/√2
    = ((|detector1> + |detector2>) - i(-|detector1> + |detector2>))/2
    = (|detector1> + |detector2> + i|detector1> - i|detector2>)/2
    = (1+i|detector1> + 1-i|detector2>)/2
    = 0.5+0.5i|detector1> + 0.5-0.5i|detector2>

  P(|detector1>) = |0.5+0.5i|² = 0.5 = 50%

  P(|detector2>) = |0.5-0.5i|² = 0.5 = 50%

One final interesting effect occurs when two separate (but otherwise indistinguishable) photons are directed into each port of the first beam splitter simultaneously. There are four possible combinations: (1) the front port photon is reflected and the rear port photon is transmitted, (2) both photons are reflected, (3) both photons are transmitted and (4) the front port photon is transmitted and the rear port photon is reflected. This results in a superposition of four states as represented below:

  1/√2(-|greengreen> - |greenred> + |redgreen> + |redred>)

However the two states with a photon on each path are actually the same state since the photons are indistinguishable. And, since their amplitudes sum to 0, they destructively interfere. Thus, if a measurement were performed at this point by adding detectors, there would be an equal probability of finding either two photons on the green path or two photons on the red path but never one photon on each path.^[6]

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[1] The complex plane is analogous to a clock face or compass face. The real number line is horizontal, with 3pm or East representing the number 1 and 9pm or West representing the number -1. The imaginary number line is vertical, with 12am or North representing the imaginary number i and 6am or South representing the imaginary number -i. The origin is the center point which is 0.

[2] The global photon in superposition could be considered an abstraction similar to a university that is an abstraction over its distinct buildings or campuses.

[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] This is known as the Hadamard matrix. It also represents the Hadamard gate in quantum computing which can be used to transform a qubit into a superposition state.

[5] The vector can be regarded as a qubit that has been prepared in state |1> (i.e., by directing the photon towards the front port of the beam splitter). The beam splitter transforms the qubit in state |1> into the superposition 1/√2(|0> - |1>) which later results in the selection of both ports of the second beam splitter.

[6] This is the Hong-Ou-Mandel effect.

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.

Interpreting Quantum Mechanics

In my last post, I discussed the mathematical intuition underlying Quantum Mechanics, which is the idea that particle configurations with probability amplitudes can cancel out when combined. That is because amplitudes can be positive, negative or complex numbers, not just positive numbers as classical probabilities are.

I also noted that when a photon arrives at the back screen in the double-slit experiment, we only see one of the slit detectors activated, which corresponds to one of the configurations. But how do we account for the configuration where the other detector was activated?

There are several interpretations, all disturbing, so therefore best described by reference to superheroes.

Copenhagen Interpretation: You, the conscious observer, create reality. If you look at the back screen, the photon will be a wave. If you look at the slits, the photon will instead be a particle. You don't feel like you're in a superposition, so you're an exception to the laws that those little particles follow.




De Broglie–Bohm (or pilot-wave) Interpretation: The photon surfs on a wave which carries it through one of the slits to a location on the back screen where destructive interference doesn't occur. Added bonus: Entangled particles can communicate faster than the speed of light in violation of special relativity.

Many-Worlds Interpretation: You observe one of the photons going through one slit while your twin observes the other photon going through the other slit. You are either observing a superposition on the back screen or participating in a branch of one when you become entangled with one of the photons. Also the entire universe is in a superposition. Things just look classical because you have to be standing in one configuration or another.





Instrumentalist Interpretation: Yada yada yada ... who cares? You can build really amazing stuff using quantum mechanics! Also affectionately called, "shut up and calculate!".

So there we have it. The Copenhagen Interpretation posits an observer-dependent reality and a mysterious wave function collapse. The Bohmian Interpretation requires information to travel faster than light.^[1] The Many-Worlds Interpretation causes incredulous stares. And, finally, the Instrumentalist Interpretation isn't an interpretation at all.^[2]

Since each interpretation uses the same mathematical formalism, is there any reason to prefer one to another on philosophical grounds? I think there is.

The famous Schrodinger's Cat thought experiment vividly demonstrates the logic of quantum behavior in terms of familiar, everyday things. The upshot is that the cat is in a superposition of being both dead and alive until we look in the box. But what does that mean? And why don't we normally observe such things?

To take the second question first, we actually can observe superpositions involving objects that are (just barely) visible to the naked eye. For example, a recent experiment demonstrated interference effects for the superposition of a tiny tuning fork vibrating and not vibrating. This is analogous to the double-slit experiment where photon amplitude flows through both slits and we observe an interference pattern on the back screen.

So what does this mean? It means that we have observed the effects of a single amplitude, and it is the sum of the amplitude for a vibrating tuning fork and the amplitude for a non-vibrating tuning fork.

How can this observation be explained in a coherent way? The idea that there is a single tuning fork that is both vibrating and not vibrating is a contradiction, so that fails. The idea that there is a single tuning fork that is either vibrating or not vibrating also fails, since possibilities can't cause interference effects. That leaves the idea that there is one tuning fork that vibrates and a second one that does not and that we are observing the combined effects of both.

If there are two tuning forks, what would explain the observation of a single tuning fork (that is either vibrating or not) when you try to detect the vibration? There are paths that entangle you with the vibrating tuning fork and paths that entangle you with the non-vibrating tuning fork and amplitude flows along both paths. Becoming entangled with one of the tuning forks destroys the interference pattern from your point-of-view (that is, you can no longer observe the effects of the other tuning fork because there are no amplitude paths from here to there).^[3]

The Many-World's Interpretation, despite its startling implications, seems to me to be the most coherent interpretation of both our everyday and quantum observations. That, of course, doesn't mean that it is true - that is ultimately an empirical question^[4]. However it is an intuitive and natural framework for conceptualizing our observed experience.

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[1] See Bells Theorem which rules out local hidden variables.

[2] The pragmatic and reasonable version of the Instrumentalist Interpretation is that the math works, whether or not we know what it means. In stronger versions, it asserts that there is no explanation to be had and that the math doesn't mean anything.

[3] This process is called decoherence and is only reversible in practice in microscopic environments where interactions with air and apparatus molecules can be controlled (it's easy to break an egg but difficult to put it back together again). So observing things in a single (or decohered) state is the everyday situation that corresponds to our classical intuitions. The difficult challenge for quantum computing just is how to reliably maintain quantum bits in a superposition of 1 and 0 values.

[4] Scott Aaronson has a great post explaining that while the Many-Worlds Interpretation is the obvious, straightforward reading of quantum mechanics, it is also provisional in a way that heliocentrism (as opposed to geocentrism) isn't. You could, in principle, fly a spaceship above the plane of the solar system and see the Earth and the other planets circling the sun. However you can't similarly travel to another world branch to meet your twin.

Visualizing Quantum Mechanics

Diagram 1: Wave-like interference pattern

In 1803, polymath Thomas Young performed the famous double-slit experiment demonstrating that light exhibits wave-like behavior. When light is shone on a plate pierced by two parallel slits, an interference pattern appears on the screen behind the plate. No light appears at those locations where the wave crests and troughs combine and cancel out, as shown in Diagram 1. Also, the light is most intense at the center of the screen where the waves combine constructively.

In more recent double-slit experiments it has been shown that the same interference pattern emerges even when light is emitted one photon at a time! How can this be? Surely each single photon must go through either one slit or the other, but then why wouldn't that produce the two-clumps pattern^[1] shown in Diagram 2 below? Perhaps an experiment could be done to observe whether the photon really does pass through one slit or the other.


Diagram 2: Particle-like versus wave-like behavior

So when detectors are placed at the slits, each emitted photon is detected passing through just one of the slits, just as we would expect, and not through both slits as a wave would. However the interference pattern then disappears!

Strangely, the photon exhibits particle-like behavior when we detect which slit it passes through but exhibits wave-like behavior when we don't try to detect which slit it passes through. Does the act of observation determine the behavior of the photon?^[2]

These observations and questions highlight the generally-perceived mystery around quantum mechanics. The purpose of this post is to describe a way to visually conceptualize what is going on in the double-slit experiment using the correct mathematical intuition. This can help us to think more clearly about these questions. So let's get started!

Consider a photon that is emitted toward the plate in Diagram 2 above. Let's also consider a single location on the back screen, labeled B0, where destructive interference would occur. There are two paths from the photon emitter to B0, one through each slit, labeled S1 and S2. A distinct state that a photon and experimental apparatus can be in (such as their positions) is called a configuration^[3].

A configuration has a value associated with it called an amplitude^[4]. An amplitude is expressed as a complex number in the form (a + bi) and can be visualized as an arrow that can point in any compass direction. Amplitude flows from prior configurations to subsequent configurations. For each configuration, the incoming amplitudes are summed. The amplitude is also transformed by rules, such as when a photon moves, changes direction through a slit or activates a detector.

Starting with the initial configuration (and amplitude) for our double-slit thought experiment, we can transition to subsequent configurations, follow the rules that transform amplitude, and see where we end up.

The configurations (including the amplitude transformation rules)^[5] are:

1. Initial configuration: (-1 - i) [arrow pointing south-west]
2. A photon goes from the emitter to S1: multiply by -1 = (1 + i) [north-east]
3. A photon goes from S1 to B0: multiply by -i = (1 - i) [south-east]
4. A photon goes from the emitter to S2: multiply by (0.5 + 0.5i) = (0 - i) [south]
5. A photon goes from S2 to B0: multiply by (-1 - i) = (-1 + i) [north-west]
6. A photon arrives at B0: (0 + 0i) [no arrow]

The total amplitude flowing toward B0 is the sum of the individual amplitudes flowing toward B0 (underlined), which is (1 - i) + (-1 + i) = (0 + 0i). The probability of a photon arriving at B0 is the squared modulus of the amplitude (a² + b²)^[6], which is 0² + 0² = 0%. Therefore no photon will arrive at B0, due to the individual amplitudes canceling each other out (i.e., the equal-length arrows pointing in opposite directions). This corresponds to the wave-like behavior that is observed when a series of emitted photons create an interference pattern.^[7]

Now consider a second experiment where photon detectors are added at slits S1 and S2, labeled D1 and D2 respectively. They will turn from off to on if they detect a photon passing through their slit. The rules are the same as for the first experiment. The configurations are:

1. Initial configuration: (-1 - i) [arrow pointing south-west]
2. A photon goes from the emitter to S1 and D1 is off and D2 is off: (1 + i) [north-east]
3. A photon goes from S1 to B0 and D1 is on and D2 is off: (1 - i) [south-east]
4. A photon goes from the emitter to S2 and D1 is off and D2 is off: (0 - i) [south]
5. A photon goes from S2 to B0 and D1 is off and D2 is on: (-1 + i) [north-west]
6. A photon arrives at B0 and D1 is on and D2 is off: (1 - i) [south-east]
7. A photon arrives at B0 and D1 is off and D2 is on: (-1 + i) [north-west]

In this experiment, the individual amplitudes flowing toward B0 (underlined) are flowing to two distinct configurations (since there can be no single configuration where D1 is both on and off), so the individual amplitudes are not summed. There is now a positive probability that the photon arrives at B0, with an equal probability of the photon being detected at either slit (the squared modulus of the amplitude for each final configuration is 2, so the ratio is 2:2). This corresponds to the classic particle-like behavior that is observed when the photon is detected going through one of the slits.^[8]

So complex addition of destination configuration amplitudes is the mathematical basis for our observations in the double-slit experiment. The classical intuition is that adding more paths to a destination makes it more likely to reach the destination. The quantum intuition is that adding more paths to a destination can make the destination unreachable since paths can destructively interfere.

This still leaves one more puzzling question. When a photon arrives at B0, we only see one of the detectors activated, which corresponds to one of the configurations. But how do we account for the configuration where the other detector was activated? I will leave this question for a future post.

The ideas presented here were inspired by Eliezer Yudkowsky's post on configurations and amplitude from his series on quantum physics. For my earlier posts on visualizing complex numbers, see Seeing complex numbers and Visualizing Euler's Identity.

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[1] To visualize photons exhibiting particle-like behavior, imagine someone with a gun firing bullets at the plate. For the bullets that pass through the slits, one clump of bullets would accumulate behind the first slit and a second clump of bullets would accumulate behind the second slit.

[2] That is, how should these counter-intuitive observations be interpreted? For one example, theoretical physicist John Archibald Wheeler once commented, "Actually, quantum phenomena are neither waves nor particles but are intrinsically undefined until the moment they are measured. In a sense, the British philosopher Bishop Berkeley was right when he asserted two centuries ago 'to be is to be perceived.'" - Scientific American, July 1992, p. 75

[3] A configuration (quantum state) is a distinct state that the universe is in at a point in time. In reality, it includes all the particles in the universe and all the particles that the emitter, plate, detectors and human observers consist of. What distinguishes one configuration from another is that at least one particle has a different property or position. When two configurations interfere, they combine to form a single configuration - an instance of the superposition principle.


Diagram 3: (a) Rotation to sine wave (b) 180^o phase shift

[4] The amplitude of a wave is the magnitude from rest to crest and is a real number - see Diagram 3. The configuration amplitude, which is the sense used here (and is elsewhere termed a probability amplitude), is a complex number (or phase vector) which additionally encapsulates a phase angle and can be visually represented as a radial arrow that points in any compass direction. For example, (0 + i) represents an anti-clockwise rotation of 90^o from 1 on the real number line and corresponds to an arrow pointing north (the wave peak). Combining two similar waves that are phase-shifted by 180^o (equal-length arrows pointing in opposite directions) results in wave cancellation (destructive interference).

[5] The initial configuration amplitude and the rules in the thought experiment are hypothetical, but serve to demonstrate the key conceptual point of amplitude interference. For the first photon path to B0, (-1 - i) * -1 * -i = (1 - i). For the second photon path to B0, (-1 - i) * (0.5 + 0.5i) * (-1 -i) = (-1 + i). These two final amplitudes have the same magnitude but are 180^o out of phase and therefore cancel out when in a superposition.

[6] This is known as the Born rule.

[7] We can also consider the center of the back screen where the light is most intense. In this case, two incoming configuration amplitudes with the same phase angle are constructively interfering, thus summing their magnitudes. For example, (1 - i) + (1 - i) = (2 - 2i) which corresponds to an arrow sqrt(2² + 2²) units in length pointing south-east.

[8] The detector (with an on or off state) could equally be replaced by a rock (that is or is not perturbed by a photon) and the computational logic would be the same.

Visualizing Euler's Identity

Euler's Identity is often considered the most beautiful equation in mathematics because it elegantly combines one each of 0, 1, i, e, π, addition, multiplication and exponentiation. But intuitively, what does it mean?

Answering that requires a brief tour of exponential growth (to understand e) and complex numbers (to understand i and π). My previous post on complex numbers is here.

e, or Euler's number, is a mathematical constant (approximately 2.72) that represents 100% continuous growth when starting at 1. For example, 100% compound interest on $1 for a year would be e in dollars, or $2.72. With continuous growth, the interest is calculated at every instant as compared with yearly (where you would only end up with $2), quarterly ($2.44), monthly ($2.61) or daily ($2.71). The formula for calculating compound interest is:

  (1) compound interest = (1 + 1 / time periods)time periods

And e is the limit that is approached as the number of time periods increase.

e is used to calculate continuous growth for any rate, time period or starting point. So 10% compound interest on $1,000 for 3 years would be $1,000 * e ^{0.1 * 3} = $1,349.86. The formula is:

  (2) final amount = initial amount * e rate * time periods

If we know the initial and final amounts but not the interest rate, then the natural logarithm function is used, as follows:

  (3) rate = ln(final amount / starting amount)

The natural logarithm of a number is the exponent that e is raised to in order to get that same number. For example, 2 = e^0.69, so the natural logarithm of 2 is 0.69. That is, the compound interest rate required to grow from $1 to $2 in 1 year is 69%.

This is where I think things get interesting! Any number can be interpreted as the end result of continuous growth starting from 1, that is:

  (4) number = e ln(number)

This same idea can be applied to numbers with exponents. For example, 2³ can be understood as 1 growing to 2 at a rate of ln(2) (or 69%) and then again to 4 and finally to 8, for a total of 3 growth periods. That is, 2³ = e ^{ln(2) * 3}. Generalizing, we get:

  (5) number time periods = e ln(number) * time periods

In geometric terms, continuous growth can be visualized as a scaling operation on the real number line (i.e., the distance travelled in each growth period is increasingly larger, as with 1, 2, 4, 8, 16, ...) However if a logarithmic scale is used for the number line then the distance travelled in each growth period will be the same (the equidistant points can be marked as 1, e, e², e³, ...)

As with real numbers, imaginary numbers also can be interpreted as the end result of continuous growth starting from 1. However instead of exponential growth along the real number line, imaginary growth follows a linear circular path around the origin on the complex plane. The reason the growth is linear is because the scale is logarithmic with a growth rate of i represented by an angular distance of 1 radian. You can see this linearity when multiplying by i. 1 multiplied by i rotates 1 by π / 2 radians (90 degrees) to i and multiplying by i again rotates a further π / 2 radians to -1.

To summarize, a growth rate of 1 (or 100%) results in growth from 1 to e¹ on the real number line, whereas a growth rate of i results in growth from 1 to eⁱ on the complex number plane - a distance of 1 radian around the unit circle (see the diagram below).

Similarly, a growth rate of 2i will travel 2 radians from 1 to e²ⁱ. A growth rate of π / 2i will travel π / 2 radians to e ^{π / 2 i} or i. A growth rate of π i will travel π radians to e^{π i} or -1. And a growth rate of 2π i will travel all the way around the unit circle to arrive back at 1.

That second-to-last calculation is Euler's Identity: e ^{π i} = -1. It just means that starting at 1 and growing at an imaginary rate of π (via an anti-clockwise rotation), we will end up at -1 on the real number line. A way to remember this is that if you deposit $1 in a bank offering π imaginary interest, you'll end up owing them $1 after 1 year! Fortunately, if you wait for another year, you will get your original money back...

The basic formulas for calculating real growth can easily be applied to imaginary numbers. The key insight is to transform the number to base e and then the growth rate will be expressed in the exponent (as imaginary or real). Some further fun equations to end this post.

  (a) 23i = e ln(2) * 3i = e 0.69 * 3i = 2.08 radians

Start at 1 and grow at a 69% compound rate three times for a distance of 2.08 radians.

  (b) i = e ln(i) = e π / 2i = π / 2 radians

Start at 1 and grow at a rate of π / 2 radians to arrive at i.

  (c) i2 = e ln(i) * 2 = e π / 2i * 2 = e π i = π radians = -1

Euler's Identity derived from the complex number identity i ² = -1.

  (d) ii = e ln(i) * i = e π / 2 * i * i = e π / 2 * -1 = e -π / 2 = 0.21

Start at 1 and grow at a rate of -π / 2 to arrive at 0.21.

  (e) (ii)i = e ln(ii) * i = e ln(e -π / 2) * i = e -π / 2i = -i

Start at 1 and grow at a rate of -π / 2 radians to arrive at -i.

Seeing complex numbers

A number line is a picture that represents numbers as points on a straight horizontal line. As can be seen in the image, the numbers increase as you move to the right along the line and decrease as you move to the left. Zero is represented in the middle of the line, bisecting the positive and negative numbers. Fractions and irrational numbers are also representable on this line between the whole number points.

Operations such as addition, subtraction, multiplication and division are simply transformations of a number on this line. For example, to add 2 to the number 3 means to start at the number three point and then move two positions to the right along the line (in geometrical terms, this is called translation). To multiply 5 by 2 is to start at the number five point and then move right until you are twice the distance from the origin (geometrically, this is called scaling).

In the sixteenth century, a new kind of number called a complex number was introduced by Italian mathematician Gerolamo Cardano. It is expressed as a + bi where a and b are real numbers and i is termed the imaginary unit which satisfies the equation i ² = -1. As negative numbers have applications in areas such as temperature measurement and finance, so complex numbers apply in areas such as physics and fractal geometry.

But is it possible to have an intuitive understanding of complex numbers? Didn't we learn in school that any number (including a negative number) multiplied by itself is always a positive number? So how can i ² = -1 make any sense? What could i be?

The first step to understanding complex numbers is to extend our concept of the number line to a number plane by adding a second dimension that cuts through the number line origin. As shown in the image, this is called the imaginary dimension with i as the unit and it is perpendicular to the real dimension.

The second step is to introduce rotation around the origin as a new transformation operation. When we rotate a number on the real number line by 90^o in an anti-clockwise direction, we can see that it ends up on the imaginary number line. This operation is what it means to multiply by i. If we rotate a further 90^o in an anti-clockwise direction (that is, multiply by i again), then we end up back on the real number line. But we are now on the opposite side of the origin. That is, if we start at 3, multiply by i, then multiply by i again, we end up at -3. That is, 3i ² = -3. The effect of multiplying by i ² is simply to negate our original number by rotating 180^o to the other side of the origin. If we multiply by i ² again, then we pass through -3i and end up back where we started at 3.

Transforming a negative number into a positive number requires translation on the real number line. Transforming a complex number into a real number requires rotation on the real-imaginary plane. So what is i? The real number one rotated by 90^o onto the imaginary number line. That's all it means.