Computer Science TopicsQuantum Computing

[Quantum Objects] Spin Class

A spinning neutron with a magnetic moment S couples with a magnetic field much like a tiny magnet. Depending on its distance from the apparatus, the magnetic field \textbf{B} it feels will change. The interaction between the magnet and the magnetic field also depends on which direction the magnet is pointing; some orientations lead to an upwards deflection, while others cause downwards deflection.

The interaction energy between a spinning neutron or a tiny bar magnet and a magnetic field is given by U = - \mathbf{S} \cdot \textbf{B}. Since our apparatus is designed so that the magnetic field \textbf{B} is only along the z-axis, the energy is simply given by U = -\mathbf{S} \cdot \mathbf{B_{z}}.

How would the deflection force caused by this interaction change if a tiny bar magnet (or a spinning neutron) were flipped 180^{\circ} from pointing “up” to pointing “down”?


Correct answer: An equal deflection force in the opposite direction

The interaction energy U between the spinning neutron and the magnetic field creates a force which acts on the neutron.

The interaction energy is given by U = -\mathbf{S} \cdot \mathbf{B_{z}}​. The dot product between the spin axis and the applied magnetic field creates a dependence on the angle between these two vectors: U = -\lvert \mathbf{S} \rvert \lvert \mathbf{B_{z}}\rvert \cos{\theta}. Rotating the angle between them by 180^{\circ} causes the sign of the interaction energy to reverse. Intuitively, we can probably guess that reversing the sign of the interaction energy would also reverse the sign of the force it causes.

From classical physics, we’re familiar with Newton’s second law that a force F can accelerate a mass m with acceleration a. On a classical particle, a force does work on the particle it is accelerating by accelerating an object a distance of d: W = F \cdot d. Work is just another word for the energy U which is put into a system. So we can represent a force as F = -\frac{U}{d}.​ So we confirm our intuition that reversing the sign of the interaction energy would result in an equal and opposite force.

The force experienced by the neutron depends on the direction of its spin moment since the dot product is equivalent to F = \frac{\partial }{\partial z} \lvert\mathbf{B}\rvert \lvert \mathbf{S}\rvert \cos{(\theta)},where \theta is the angle between the magnetic field and the neutron spin axis.

If the axis of spin \mathbf{S} is in the same direction as \mathbf{B}, then the dot product is positive and the force pushes the neutron into regions of higher magnetic field. However, if \mathbf{S} is in the opposite direction as \mathbf{B}, then the dot product is negative, pushing the neutron in the opposite direction. If the neutron just happens to be directly aligned with the z-axis of the magnetic field, it would be maximally deflected, while one pointed directly perpendicular (along the xy-plane) would not be deflected at all.

If we place a particle detector on the opposite side of the magnet, we can measure the magnitude of the neutron deflection, giving us an indication of which direction its spin is directed.

Just as soon as we’ve got the basics down for the behavior of a single neutron, another neutron emerges from the thermonuclear oven, and appears to have the same initial ballistic trajectory as the first.

How do you expect this new neutron to be deflected by the magnet apparatus?


Correct answer: The deflection will be random

We haven’t specified any details about the nuclear oven which is generating our neutrons for this experiment, other than to say that it is fueled by nuclear reactions, and is extremely hot. Even if this oven was initially prepared with oriented nuclear fuel, the heat and random motion in the oven would lead to any produced neutrons having random orientations.

Temperature is just a measure of how much energy is contained in each particle on average. This energy can go into spinning, translating (moving), or vibrating particles. At extremely cold temperatures, particles can stay still relative to one another, since they don’t have enough energy to move or rotate around. At even moderately warm temperatures like room temperature, particles as small as atoms and molecules have so much energy that they move, vibrate, and rotate freely. Collisions between particles rapidly lead to random thermal motion, which would surely be present in any nuclear oven.

For this reason, we’d expect a random orientation for each neutron that emerges from the oven, and if the orientation is random, then the deflection force will also be random in magnitude and direction.

Let’s turn up the heat in the thermonuclear oven generating our neutrons. Now we’re generating a constant stream of neutrons which emerge from the tiny aperture to our oven pointed in the same direction, headed towards the magnetic field and the detector beyond.

Our neutron analyzer measures deflection of neutrons using a magnetic field, with the deflection force depending on the direction of spin. By observing the neutron deflection, this analyzer is really measuring \mathbf{S_z}​, the part of the spin axis which aligns with the applied magnetic field in the z-direction.

What would we expect the distribution of a constant stream of neutrons to look like on our detector screen?


Correct answer: A continuous distribution centered around zero deflection

We are assuming that the neutrons emerging from our nuclear oven would be oriented in completely random directions in space. But our z-aligned magnetic field is only deflecting the neutrons based on the z-component of their spin vector. A neutron aligned along the positive z-direction would be maximally deflected in the “up” direction, while one aligned along the negative z-direction would be maximally deflected in the “down” direction. A neutron with a spin vector aligned completely on the xy-plane (perpendicular to the z-axis) would not be deflected at all.

The likelihood of a randomly oriented neutron to be aligned along the positive and negative z-axes is identical, so we would certainly expect the distribution of neutrons on the detector screen to be symmetric about zero deflection. But there are far many more possible directions on the xy-plane than along either of the z-directions.

We can conclude that the distribution will be continuous (there is an infinite number of possible alignment directions), and will be centered with maximal likelihood at zero deflection.

A classical neutron can point in any direction at all, with no preferred orientation. Randomly oriented neutrons would be broadly spread across the detector, centered at no deflection.

Neutron spin is not behaving like a classical property:

We don’t see a broad spread of randomly oriented neutrons. We see two beams: one oriented up on the z-axis, and one oriented down. According to our analyzer, neutrons are only found in two possible states out of the infinite number of random orientations they could have.

Spin must be a quantum property because it is quantized, meaning it is only allowed to have certain values. Please leave your classical expectations at the door.

Why is this happening? What’s so special about neutrons that they have to be quantum while bar magnets and baseballs happily follow classical expectations?

Well, we chose neutrons for our case study because classically they shouldn’t be affected at all by a magnetic field. But they are deflected in this strange binary way because they have a quantum property that we call spin. It’s important to note that the neutron isn’t spinning—at least not in any classically meaningful sense. Spin is a property that every particle has, but its effects are very small.

At the human scale, objects are composed of extraordinarily large numbers of microscopic quantum objects; including neutrons, protons, and electrons. Why don’t we observe the binary quantum effects of spin in macroscopic objects?


Correct answer: Quantum effects fluctuate and tend to average out in macroscopic objects

Macroscopic objects at the human scale contain on the order of 10^{24} microscopic quantum objects, including protons, neutrons, and electrons. Each of these quantum objects has an intrinsic spin direction, and in isolation, each object would behave in the strange binary manner that we’ve observed here.

But when quantum objects are combined in large numbers, their spin axes are randomly oriented and completely uncorrelated with one another. In a magnetic field, each spin might cause a small force along its axis, but when averaged over every random orientation, the effects tend to cancel out. Only in special materials or very cold objects that don’t have the random orientations caused by thermal noise can spins be aligned in bulk and cause an observable macroscopic effect.

At the human scale, binary quantum effects like spin are averaged over incredibly large numbers of microscopic objects, and tend to smooth out to reproduce continuous classical behavior.

Only extremely precise measurement of isolated quantum objects can observe the effects of spin. The strength of the neutron spin magnetic moment is about 10^{34} times lower than the strength of bar magnets you’re used to. This is why quantum mechanics was discovered by a bunch of tenacious physicists who were obsessed with the details past the sixth place of decimals.

Particle spin just smashed our classical expectations.

In classical physics, the value of physical quantities like direction of spin, momentum, and position can vary continuously. There are no rules that allow or disallow certain values.

But the quantum property we call spin behaves differently. A nuclear oven should produce neutrons in any of an infinite number of possible orientations, but it turns out there are only ever two: spinning on an axis aligned up, or spinning on an axis aligned down.

Spin is our first encounter with a purely quantum property, and it was first observed experimentally using silver atoms produced in a furnace and a similar magnetic apparatus by two physicists in 1922. Since then it’s been observed in every type of microscopic particle, including neutrons. The experimental setup we’ll be using is still named after the original experimenters: Otto Stern and Walter Gerlach.

We’ve already seen that our knowledge of classical mechanics gives us limited predictive ability, so we will proceed cautiously into the dark.

The Stern-Gerlach (SG) analyzer will be a workhorse in this chapter, so let’s take a moment to clarify its operation. Stern and Gerlach’s experimental device was a bit different, but we’ll be able to extract all the quantum rules we need from this simple version.

The analyzer has an input for incoming particles, and only two possible output channels split by an applied magnetic field: up and down. We’ll label the analyzer Z for the axis along which the magnetic field \mathbf{B} is oriented.

The strange brackets around ↑ and ↓ are called “kets” and are simply a way of referring to the state of the particles in each channel. ∣↑⟩ particles have been measured to have their spin vector aligned with the positive z-axis, and ∣↓⟩ particles were aligned with the negative z-axis. The state of the particles in the input beam is unknown, so we can’t assign them a state ket like ∣↑⟩ or ∣↓⟩. We instead refer to them using a mystery ket called ∣ψ⟩.

In the first quiz, we chose the z-axis along which to measure the spin component; but there is nothing sacred about this direction in space, as we’ll soon see. We could have chosen any other axis and would have obtained the same startling binary result.

After picking their jaws up off the floor, and recovering the contents of their minds from around the room, the surprised Stern and Gerlach returned to the startling binary result on their analyzer to learn more. We shall do the same.

Suppose the SG analyzer has now been running for quite a while, and a very large number of neutrons had been sorted into ∣↑⟩ and ∣↓⟩ states.

Based on your understanding of classical physics, how do you expect the neutrons’ states to be distributed between ∣↑⟩ and ∣↓⟩?


Correct answer: A roughly even 50:50 distribution

Though the behavior of the neutrons in the Stern-Gerlach analyzer is certainly bizarre, we still have no reason to assume that any orientation is preferred coming out of the nuclear oven. Since the oven is so hot, all of the particles inside must be involved in random thermal motion that would make any orientation equally likely.

Even though only two quantum orientations are measured instead of the infinite number of classical orientations that should be possible, we must still expect a random distribution. So for a large number of neutrons, there will be a roughly 50:50 split between the two orientations.

The number of neutrons in each channel of the analyzer is probabilistic: there is a 50:50 chance for each neutron that enters the analyzer to emerge from each output.

Recall that based on our understanding of classical mechanics, we expected the distribution of neutron spins from our thermonuclear oven to be randomly oriented due to thermal motion in the oven. In classical mechanics, there is no preferred orientation. A lot seem to be different with quantum objects, but some remain the same. Though only two states of quantum spin are allowed, this expectation of random orientation remains.

This early correspondence between quantum and classical mechanics hints at a connection that will be seen time and again throughout this course. What quantitative conclusion can we make from comparing the classically expected and quantum distributions?


Correct answer: The average result ⟨x⟩ of many quantum events will match with the classical result

The behavior of quantum objects such as the spin deflection d may wildly diverge from our classical expectations, but the average ⟨d⟩ over many experiments is always the same as the classical result.

Probabilistic behavior is at the heart of quantum mechanics. Unsurprisingly, this behavior is one of the problems that many early theoretical and experimental physicists alike had with the theory. Many physicists argued that since we cannot predict from which output a given neutron will exit, there must simply be variables that remain unknown to us. They assumed that given complete knowledge of the system, we’d be able to deterministically predict the path of each neutron.

After much hemming and hawing over the past century, most physicists have decided that a certain amount of unpredictability is a fact of nature, though this point remains one of the biggest questions at the foundations of quantum mechanics.

Let’s extend the experiment a bit further.

Suppose we place a second SG analyzer with its input aligned with the ∣↑⟩ channel of the first analyzer. So after observing the neutrons to have a spin axis aligned with the positive z-axis and assigning them the state ∣↑⟩, the neutrons proceed directly into a second SG analyzer which performs an identical measurement.

What do you expect the distribution of neutrons to look like emerging from the ∣↑⟩ and ∣↓⟩ exits of the second analyzer?


Correct answer: All neutrons emerge from the  ∣↑⟩ channel

First let’s walk through the observations we’ve seen so far. The neutrons emerging from the nuclear oven appear to be randomly oriented, even though there are only two allowed quantum spin directions. The first Stern-Gerlach device filters these neutrons into two new beams: one consisting of neutrons that have been measured to be in the ∣↑⟩ direction, and one for neutrons that have been measured to be in the ∣↓⟩ direction.

By placing another Stern-Gerlach analyzer in the path of the ∣↑⟩ neutrons, we are essentially repeating this filtering event. Since all of the neutrons are known to be in the ∣↑⟩ direction, the neutrons are only deflected in the “up” direction as expected for that direction, so all emerge from the ∣↑⟩ exit.

We can all breathe a small sigh of relief: the state of the neutrons which emerge from the ∣↑⟩ channel of the first analyzer remains consistently in the ∣↑⟩ channel through the second analyzer. Though the first measurement of the unknown ∣ψ⟩ neutrons seemed to be completely random and binary, once the neutrons have been observed in one state, they appear to remain in it.

Indeed, this result doesn’t just hold for two SG analyzers in series: the same result will be true for any number of repetitions. Given no other interfering magnetic fields or forces, the ∣↑⟩ state which is prepared by the first SG analyzer is robust to subsequent measurement by other analyzers.

Our physical understanding of the ∣↑⟩ neutron channel is quite simple: these neutrons have had the z-component of their spin axis measured, and they were found to align exactly with the magnetic field along the z-axis. These particles are usually referred to as having “spin up”, while the ∣↓⟩ neutrons are referred to as having “spin down”.

But spin is a quantum property. Do we actually think that these particles are spinning like tops as they proceed through the magnetic fields of the SG devices? Stern and Gerlach had the same misgivings about referring to this quantum property as spin, so they devised a very simple alteration to their experiment.

If this model of spinning neutrons is correct, what channel would you expect the neutrons to emerge from if we flipped the magnetic field in the second SG device upside-down?


Correct answer: All neutrons emerge from the “down” channel

If the physical model of neutrons acting like microscopic spinning tops is correct, then flipping over the magnetic apparatus in the Stern-Gerlach analyzer should flip the orientation of the inhomogeneous magnetic field, reversing the orientation of the interaction energy landscape, and reversing the sign of the gradient of the magnetic field. The deflection force F is given by F \propto \frac{\partial B}{\partial z}. Flipping the orientation of the magnetic field B would be expected to reverse the direction of the deflection force. ∣↑⟩ neutrons would normally be deflected “up” by the apparatus, but will be reflected “down” in this inverted apparatus and emerge from the ∣↓⟩ exit.

This leaves us with a physical model that is consistent with a neutron beam acting as a stream of subatomic spinning tops, each of which creates a magnetic moment along their axis of spin just like a classical vector quantity.

This physical insight into the nature of spin gained from flipping over an experimental apparatus inspired Stern and Gerlach: what if we rotated the magnetic field by a different angle?

In physics, an observable is a dynamic variable describing a state which can be measured. From a spinning top, we could observe the x-, y-, and z-components of the spin axis to find the spin vector. The angular velocity and kinetic energy would also be physical observables associated with the spin.

The easiest way to measure the direction of the spin vector might be to rotate your perspective. You could capture the orientation by taking a photo of the top from the three cardinal directions: one from above looking perpendicular to the xy-plane, and two others directed at the xz- and yz-planes. From these points of view, we can measure the x-y-, and z-components of the spin vector and fully describe the orientation of the top.

We’ve had some success understanding neutrons as microscopic spinning tops, even though a spin vector that can only point up or down is an odd vector indeed. Let’s see if we can measure the other neutron spin components with an SG analyzer by rotating our perspective.

So far, the particle spin axis seems to be acting much like a spatial vector along the z-direction, since it reverses direction when we flip our magnetic field aligned along that axis. The ∣↑⟩ and ∣↓⟩ states behave like spinning neutrons aligned along the positive and negative z-axes, respectively.

To completely describe the spin of a top, we would need to observe its spin axis from several directions and record the x-y-, and z-components.

How can we measure the x-component of the neutron spin vector using an SG analyzer?


Correct answer: Rotate the magnetic field by 90^{\circ} around the y-axis.

The z-aligned Stern-Gerlach analyzer measures the z-component of a neutron’s spin vector by measuring the deflection in a z-aligned magnetic field.

If the magnetic apparatus in a Stern-Gerlach analyzer were rotated 90 degrees, the magnetic field would be aligned along the x-axis, and a neutron’s deflection would be proportional to the x-component of the neutron’s spin vector.

Instead of flipping the second analyzer upside down, let’s rotate it by 90 degrees around the y-axis. Now the magnetic field is oriented along the positive x-axis, and we’ll refer to this device as an SG-x analyzer.

Following the terminology we learned in the last quiz, the first SG-z analyzer prepares the ∣↑⟩ and ∣↓⟩ states and filters them into two channels. Our physical model of these states suspects that they correspond to neutrons spinning around the positive and negative z-axes. The second SG-x analyzer now measures the x-component of the \mathbf{S} vector.

If our understanding of the ∣↑⟩ and ∣↓⟩ states is correct, given a classical model of the magnetic field, what would we expect to observe from the second SG-x analyzer?


Correct answer: The neutrons are not deflected at all

The z-aligned Stern-Gerlach analyzer measures the z-component of a neutron’s spin vector by measuring the deflection in a z-aligned magnetic field.

If the magnetic apparatus in a Stern-Gerlach analyzer were rotated 90 degrees, the magnetic field would be aligned along the x-axis, and a neutron’s deflection would be proportional to the x-component of the neutron’s spin vector. Since the first SG-z analyzer filters the randomly oriented neutrons from the oven into ∣↑⟩ and ∣↓⟩ which are aligned along the positive and negative z-directions, respectively, we can predict the x-component of the spin vectors we’ve already measured.

The z- and x-axes are orthogonal to each other, so any vector aligned along one of these axes has no component along any other orthogonal axis. Since we know the filtered z spin directions are aligned along the z-axis, they should not have any x-component, so we’d expect them to not be deflected in an x-aligned SG analyzer.

The importance of this prediction calls for going through its derivation in detail:

When measuring the z-component of prepared ∣↑⟩ neutrons, we expect to see maximal “up” deflection, since the neutron spins are parallel with the magnetic field. Similarly, when we flipped the second SG analyzer upside-down, the deflection reversed since the spin magnetic moment was now anti-parallel to the magnetic field.

Rotating the second SG analyzer by \ang{90}^{\circ} makes the neutron spins aligned orthogonally to the magnetic field. Since the x-component of a vector in the z-direction is zero, we’d expect the deflection force to be zero. Thus, we would expect no neutrons to emerge from either ∣↑⟩ or ∣↓⟩ exit.

This is not what we observe: we again see a binary result, half of the neutrons are deflected as though their spin were aligned along the positive x-axis, and half as though they were aligned along the negative x-axis.

The SG analyzer refuses to measure the expected 0 deflection force even when measuring along an axis orthogonal to its spin. Instead, it stubbornly produces the same binary splitting as though half of the neutrons were aligned along each direction of the x-axis, which we will refer to as ∣→⟩ and ∣←⟩.

Somehow the spin vector \mathbf{S} just got even stranger: particles observed to be aligned along the z-axis also seem to be simultaneously aligned along the x-axis.

A normal everyday vector cannot be aligned along two orthogonal axes simultaneously, but that appears to be exactly what we observe. In fact, we can repeat this measurement with other rotations of the second SG analyzer, and we continue to see the same binary result for any set or order of orthogonal measurements from which we’d expect 0 deflection force.

What physical law of quantum objects might explain these observations?


Correct answer: Physical quantities of quantum objects have certain allowed values, and 0 isn’t one of them

Quantum mechanics gets its name from the interesting observation that when it comes to microscopic phenomena, physical quantities like energy, momentum, and spin can only assume quantized values. In the case of the neutron spin system, the only allowed values are “up spin” aligned parallel to the direction of the magnetic field, and “down spin” aligned antiparallel to the magnetic field.

Even though the classically expected spin vector in this experiment would be 0 spin, this is not one of the allowed values, so the particle is still observed only in the “up” and “down” states.

Spin is a quantum property, meaning it can only be observed in a given set of quantized values, and in this case, 0 is not one of those values, even though that would be the classically expected solution.

But there is a glimmer of classical physics remaining: if we repeat this experiment many many times, the number of ∣↑⟩ neutrons which are measured to be ∣→⟩ and ∣←⟩ are equal–so the average deflection is zero, matching with the classical expectation:

At the limit of many quantum measurements, the average will always approach the classical result, even if individual experiments appear to violate the classical law. This is called the classical limit.

Let’s say we don’t rotate the second SG analyzer by 90^{\circ} from the z-axis, but instead by an arbitrary angle, \theta. If we treat the deflection of the ∣↑⟩ and ∣↓⟩ states to be +1 and -1, respectively, what would be the average deflection?

Hint: Calculate the classical limit of this experiment to find the average of many quantum measurements.


Correct answer: An average deflection of \cos{(\theta)}

We’ve learned that the average of many quantum measurements will always match with the classically expected value.

For an arbitrary rotation \theta of a second z-aligned SG analyzer, we end up with a magnetic field aligned along a new vector \vec{i} not necessarily aligned with the x- or z-axis. The deflection force of neutrons filtered into the positive z-aligned direction can be calculated using the dot product between the neutron and the magnetic field:

\begin{aligned} F &= \mu \cdot \frac{\partial B}{\partial z} \\ &= \mu \frac{\partial B}{\partial z} \cos{\theta}. \end{aligned}

If the deflection observed for the ∣↑⟩ and ∣↓⟩ directions (for \theta=0 and \theta=\pi) is +1 and -1, respectively, then the classically expected deflection force for a second SG analyzer with an angle \theta between the rotated magnetic field and the prepared spin directions would be \cos{\theta}.

The average of many quantum measurements will tend towards the classically expected deflection of \cos{\theta}.

The SG analyzer will only ever observe one of the two allowed quantized spin values. Each corresponds to maximum deflection as though the neutrons were aligned parallel or antiparallel to the magnetic field. When measuring with an SG-z analyzer, the only two observations are ∣↑⟩ and ∣↓⟩; when measuring with an SG-x analyzer, the only two observations are ∣←⟩ and ∣→⟩.

Though these are the only two allowed observations, their distribution is biased to produce a statistical average matching the classical limit. For a rotation of \theta away from the z-axis, this corresponds to an average deflection of \cos{\theta}.

The fact that the statistical average of a quantum measurement tends towards the classically expected result at the limit of many measurements explains in a large part why quantum behavior is so counterintuitive: Macroscopic objects consist of extraordinarily large numbers of microscopic particles, and we usually only observe their averages unless we’re looking very closely.

Our observations of neutron spin suggest it’s similar to a spatial vector, insofar that it flips with the orientation of the analyzer. But in other ways, it seems to be fundamentally incompatible—it’s hard to imagine a physical vector that can be aligned with two different directions in subsequent measurements.

As a sanity check, let’s add a third SG analyzer to the previous experiment, aligned like the first to measure the z-component of the neutron spin \mathbf{S} again.

Based on our results thus far, what channel(s) would you expect the neutrons to emerge from in the third analyzer?


Correct answer: All neutrons emerge from the ∣↑⟩ channel

Note that this is another time that our intuition leads us astray. All of the neutrons have been “filtered” to be in the ∣↑⟩ spin, and so we’d expect them to remain in the ∣↑⟩ spin state. But we’d be wrong—as we’ll see next.

First let’s revisit the observations we’ve seen so far: once the randomly oriented directions emerging from the nuclear oven are filtered by an SG-z analyzer, that filtering can be repeated many times, and the neutrons are still always observed to be aligned in the same “up” or “down” direction.

We’ve now measured the “left” and “right” spin direction as well, but we have no reason to expect that measuring the x-component would have any effect on the z-component. Since the experiment shows the neutrons which have been measured to be in the ∣↑⟩ are later analyzed by the successive SG-x and SG-z analyzers, we’d expect the SG-z analyzers to still be in the ∣↑⟩ direction.

We’ve already observed that the ∣↑⟩ state prepared by the SG-z analyzer is robust to multiple successive measurements. Given this understanding, we’d expect that the third measurement would observe that all the neutrons have spin up along the z-axis, because that’s the state that we prepared at the first SG analyzer.

This is not what we observe. The neutrons leaving the third SG analyzer are split with a 50:50 probability into ∣↑⟩ and ∣↓⟩, even though we diverted all ∣↓⟩ neutrons away from the experiment with the first analyzer.

How can we explain this contradictory result?


Correct answer: Measuring spin along the x-axis destroyed all previous knowledge of the neutron spin state

Our confused physical model was incorrect: we don’t have a strange vector which is aligned along two orthogonal axes at the same time. Instead, each measurement seems to reset the spin onto the axis of measurement. In effect, the last two analyzers are acting just like any other pair of orthogonal analyzers.

The initial measurement that prepared the ∣↑⟩ state is forgotten.

This shocking result demonstrates another key feature of quantum behavior:

Quantum measurements are not as gentle as classical measurements. Sometimes measuring a quantum state can change the state for all future measurements.

The unknown state ∣ψ⟩ which emerged from our thermonuclear oven was never aligned along the positive and negative z-axes. By measuring the z-component of the spin, we changed the state into one of two possibilities: ∣↑⟩ and ∣↓⟩. Measuring another component caused the same result. The SG-x analyzer reset the prepared ∣↑⟩ state into one of two completely different states: ∣→⟩ and ∣←⟩.

One might ask, “Can we be more clever in designing the experiment such that we don’t disturb the system?” No, there is a fundamental incompatibility in trying to measure the spin component of a quantum object along two different directions. So we say that the x and z spin components are incompatible observables. We can’t ever know them both simultaneously.

Let’s revisit our classical model of spinning tops. Not being able to measure both the \mathbf{S_z} and \mathbf{S_x}​ spin components is completely different than the classical case where we can measure all three components of a spin vector easily. Only after determining all three of these components can we determine the direction that the spin vector is pointing.

For quantum objects, the different spin components are incompatible. We can never know which way the neutrons are pointing. When we say “the spin is up”, we really mean only that the spin component along one axis is up. This property isn’t permanent, since measuring the spin along any other axis will reset the spin and split it back into parallel and antiparallel states.

In the last quiz, we encountered head-on the measurement problem in quantum mechanics: we could not devise an experiment capable of measuring the neutron spin components without destroying previously collected information along the way. It appears that the experimental apparatus we’re using must disturb the neutron spins while observing them, making it impossible to precisely determine the direction of the spin vector. This interesting consequence of quantum objects is not limited to spin, as we’ll see in later chapters.

How might we be disturbing the neutrons? Since this quantum measurement problem is generally only observed with microscopic systems, we might guess that our macroscopic magnets generating the magnetic field \mathbf{B} are simply too strong: it’s easy to imagine measuring the speed of a baseball with a radar gun, but difficult to imagine measuring the speed of a neutron, whether or not quantum mechanics governs its motion.

Let’s investigate this measurement problem a bit more. Can we determine what step in the SG experiment is disturbing the state of our neutrons?

Let’s revisit the experimental setup with three SG analyzers. The first SG-z analyzer filters the unknown state ∣ψ⟩ into two possibilities: ∣↑⟩ and ∣↓⟩, corresponding to “up” and “down” spin neutrons. We’ve now set up our apparatus so that 100 ∣↑⟩ neutrons are prepared by the first analyzer.

The output from this first analyzer is directed towards the input of an SG-x analyzer. Measuring the x-component of spin is incompatible with the z measurement we’ve already performed, so this analyzer resets the neutrons into two states: ∣→⟩ and ∣←⟩, aligned parallel and anti-parallel with the x-axis.

The state that an incident ∣↑⟩ neutron collapses into is probabilistic, and so with 100 neutrons input into the analyzer, we would expect 50 neutrons in the ∣→⟩ channel, and 50 with the ∣←⟩ channel.

Finally, we direct the ∣←⟩ channel into a third SG-z analyzer. Even though these neutrons have been measured previously as ∣↑⟩, that knowledge has been permanently erased by measuring an incompatible observable. Instead of confirming the ∣↑⟩ spin of all 50 or these neutrons, we again observe a reset: 25 neutrons in the ∣↑⟩ channel, and 25 in the ∣↓⟩ channel.

We’ve made a slight variation to our experimental apparatus.

Instead of measuring the ∣←⟩ channel, we’ve selected the ∣→⟩ channel and used that as input for the third analyzer.

Given what we’ve learned about incompatible measurements destroying previously collected information, what do we expect to output from the third SG device?


Correct answer: The neutrons are split evenly between both channels

We’ve learned that measuring the x-component of the neutron spin vector resets the neutron randomly into one of two possible states: ∣←⟩ and ∣→⟩. Subsequent measurement of the z-component again resets the neutron randomly into the two z states ∣↑⟩ and ∣↓⟩.

These measurements are called incompatible. It doesn’t matter which state a neutron has been measured as; subsequent measurement of an incompatible observable results in a reset into a random allowed state.

Measuring the xx-component of the spin destroys any previously collected knowledge of the spin state. So whether or not you analyze the \ket{\rightarrow}∣→⟩ or \ket{\leftarrow}∣←⟩ channels, you still have no information about the zz-component, so you get a random 50:5050:50 distribution.

Let’s now combine the \ket{\rightarrow}∣→⟩ and \ket{\leftarrow}∣←⟩ beams of neutrons which emerge from the second SG analyzer. The exact experimental method we use to perform this combination isn’t important; we use an arbitrary set of magnets to deflect the beams back into alignment.

This counterintuitive result of combining spin channels also reveals an important caveat to keep in mind when analyzing quantum systems:

The probabilistic behavior of quantum objects is not limited to SG analyzers, and it’s often tempting to break a quantum system down into simpler independent paths and combine their probabilities.

We can try to understand the combined channel experiment by analyzing two single channel experiments, and note the probability of a neutron emerging in each state:

But when we put these two results together and see what they predict for each channel of the combined experiment, we reach a glaring contradiction:

What should we conclude from these results?


Correct answer: Quantum measurements can’t be treated as independent events

If we were to use classical probability analysis, the first one-channel experiment would indicate that the total probability for the two steps through the analyzers is 25%. Likewise, the second indicates that the total probability to take the lower path through the second analyzer and output through the upper channel of the third analyzer is also 25%. Hence the total probability to output from the upper channel of the third analyzer when both paths are available would be their sum: 50%, and likewise for the channel from the lower channel.

In the single channel experiments, 50% of the neutrons are blocked after the second analyzer and 25% of the atoms exit the lower port of the third analyzer. In the combined channel experiment, 100% of the atoms pass from the second analyzer to the third analyzer, yet fewer neutrons come out of the lower port. In fact, no atoms make it through the lower port at all! This experiment allowed more ways or paths to reach the final ∣↓⟩ channel but ends up with fewer neutrons making it.

Classical probability can’t hope to explain this part of quantum mechanics.

In the single channel experiments, 50% of the neutrons are blocked after the second analyzer and 25% of the neutrons exit the ∣↓⟩ channel of the third analyzer. In the combined channel experiment, 100% of the neutrons pass from the second analyzer to the third analyzer, yet fewer neutrons come out in the ∣↓⟩ channel. In fact, there are no neutrons in that channel at all!

This combined experiment allowed the neutrons more paths to reach the final ∣↓⟩ channel but ends up with fewer neutrons making it. It’s as though we opened a second window to a room and part of the room got darker: classical probability can’t hope to explain this part of quantum mechanics known as interference, in which combining two effects can lead to cancellation, rather than enhancement. Interference will play an essential role in our later adventures with quantum objects.


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