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.