The Quantum-Classical Transition

According to the correspondence principle, in systems and events involving actions much larger than Planck’s constant the quantum-mechanical description becomes almost identical to the classical-mechanical description. That principle is a little difficult to understand on first impression because the quantum theory and classical dynamics use two different mathematical formalisms, so the correspondence principle obliges us to devise some way in which we can compare the two theories.

In Newtonian dynamics we specify the forces acting upon a particle or body, calculate the consequent acceleration of the particle or body, and solve the resulting second-order differential equation (with respect to elapsed time) to obtain an equation of motion. According well with human intuition, an equation of motion provides an algebraic description of the location of a single point, occupied by the particle or body, as a function of elapsed time. Basically it’s something we can follow with a map and a clock.

If, instead of forces, we know the energies contributed to a particle or body by surrounding systems, then we would use the Hamiltonian form of classical mechanics. In that version of classical dynamics we use the Euler-Lagrange equations, derived from the principle of least action, to generate the equations of motion or the differential equations that lead to them.

In quantum mechanics we assign to a particle a state function, which describes a composite field that fills the space around the particle. That field consists of an aleatric field (which encodes the particle’s necessary properties, such as mass, electric charge, spin, etc.) and a ghostwave (which encodes the particle’s contingent properties, such as linear momentum, kinetic energy, angular momentum, etc.). Although it conforms to the mathematical description of a forcefield and a forcewave, the composite field of quantum mechanics doesn’t actually do anything dynamic: it merely vibrates and propagates.

But Born’s theorem tells us that if we square the state function (that is, multiply it by its own complex conjugate), we obtain a function through which we calculate the probability density of finding the particle at a given point in space at a given instant of time. With that probability density we can then calculate an expectation value, a kind of average, for any of the particle’s contingent properties.

We thus have two pictures of Reality, one deterministic and one probabilistic. How do we reconcile them?

Mathematically we have a straightforward procedure. Typically the state function of a particle consists of a set of sinusoidal waves whose collective amplitude, if plotted on a graph, would resemble the bell-shaped curve of the Gaussian probability distribution. In the case of the Gaussian curve itself, for example, we have the amplitude as

(Eq’n 1)

in which h represents Planck’s constant and x has units of action. We would then have a probability density proportional to the square of that function,

(Eq’n 2)

If we integrate that function over the full volume of space, we get

(Eq’n 3)

That statement remains true to mathematics if we change the value of h; in particular, it remains true if we let h approach a value arbitrarily close to zero. In that case, as h grows smaller, the graph of the probability density grows narrower and taller until it becomes indistinguishable from an infinitely narrow, infinitely tall spike, a Dirac delta.

In essence a Dirac delta occupies a single point and thus it represents a deterministic representation of the particle described in part by Equation 1. Because the Dirac delta only has a non-zero value where its argument equals zero, the algebraic expression in that argument corresponds to the classical equation of motion. Thus we can see how a quantum description of a particle can become a classical description. Of course, we can’t change the value of Planck’s constant, but we get essentially the same result if we make the action very much larger than Planck’s constant.

So in the mathematical description the quantum-classical
transition comes smoothly and easily. In the physical description it’s not so
clear that there __is__ a quantum-classical transition. Erwin Schrödinger’s
infamous cat-in-a-box experiment still raises doubts about the validity of the
quantum theory as a correct description of Nature.

In a modified version of Schrödinger’s imaginary experiment, the apparatus consists of an unstable atom, a Geiger counter, an amplifier, a solenoid, a delicately balanced hammer, and a toggle switch that connects a battery to a light bulb through a wire. The experiment consists of the apparatus performing the following actions: the atom decays, the Geiger counter detects the decay, the amplifier turns the Geiger counter’s signal into an electric current pulse, the current causes the solenoid’s core to move and strike the hammer, the hammer falls and hits the toggle switch, and the toggle switch closes the circuit, which allows electricity to flow between the battery and the light bulb, thereby making the light bulb glow. The unstable atom is clearly a quantum system, properly described by a state function, and the hammer hitting a toggle switch to turn on a light is clearly a classical system, properly described through equations of motion. Where and how in this experiment does the quantum-classical transition take place?

To answer that question we need to take a more detailed look at the parts of the experimental apparatus and their actions.

For our unstable atom we’ll take, for example, an atom of
Magnesium-27 (_{12}Mg^{27}). With a half-life of 9.458 minutes,
Mg-27 decays into Aluminum-27 (_{13}Al^{27}) by emitting an
electron and an electron anti-neutrino. The electrons coming from Mg-27 decays
carry a mean value of 702 kilo-electron-volts in kinetic energy and that makes
them easy to detect with a Geiger-Müller counter. Assuming that the atom sits
motionless in the apparatus, we assign to it a state function that consists of
two terms,

(Eq’n 4)

in which T represents the lifetime of the magnesium atom, which is related to
the half-life t_{1/2} by T=t_{1/2}/ln2=t_{1/2}/0.693.
The first term in that state function represents the magnesium atom (with rest
mass m_{1}) and the second term represents the aluminum atom (with rest
mass m_{2}).

The exponential decay factors in that state function enable us to calculate the probabilities that the atom is either magnesium or aluminum at any given instant. We say that the atom exists as a superposition of eigenstates, one eigenstate being purely magnesium and the other being purely aluminum. But we cannot properly say that the atom is both magnesium and aluminum. Properly speaking, the atom is neither magnesium nor aluminum, but exists in an indeterminate state until it interacts with some other object or with its own parts in the beta decay, thereby producing the equivalent of the observation or measurement required to manifest an eigenstate.

The Geiger counter, which will detect the decay, consists of a Geiger-Müller tube, a DC power source, and the circuitry necessary to convert a faint pulse of electricity into a strong pulse. The Geiger-Müller tube consists of an insulated tube with a metal lining, a straight wire running down its axis, and a thin window at the free end (the opposite end has the wire connecting to the power source and the amplifying circuitry). With the tube filled with an inert gas (typically a noble gas plus a little halogen) and an electrostatic potential, usually between 400 and 600 volts, established between the anode (the straight wire running down the center of the tube) and the cathode (the tube’s metal lining), the Geiger counter stands ready to detect high-energy particles, such as those emerging from radioactive decays.

We suspend our magnesium atom inside the Geiger-Müller tube to ensure that the tube will detect its decay. At a certain instant the magnesium atom definitely turns into an aluminum atom by emitting an electron and an anti-neutrino. At that instant the state function of Equation 4 abruptly changes in a quantum jump (the subject of a separate essay) to a new state function, one with three terms. Each term describes the aleatric field of its particle (the aluminum atom, the emitted electron, or the anti-neutrino) and is multiplied by a wave function that describes the ghostwave associated with the particle. Because Al-27 is stable, the exponential decay factors don’t appear in the new state function.

The emitted electron carries between 500 and 1000 kilo-electron-volts and it takes two dozen electron-volts or less to knock an electron off one of the atoms in the gas filling the Geiger-Müller tube, so the emitted electron can conceivably ionize tens of thousands of atoms before it gets pulled into the tube’s anode. Each of the new electrons gets accelerated toward the anode by the electric field between the cathode and the anode, thereby gaining energy enough to ionize more atoms. The electron emitted from the magnesium atom thus instigates a chain reaction of ionizations, called a Townsend avalanche, that sends millions of electrons into the anode, where they form a minuscule pulse of electric current. The circuitry in the Geiger counter amplifies that pulse to a strength that, when we feed it to a loudspeaker, produces an audible click.

In our experiment the amplified pulse does not go to a loudspeaker; rather, it goes to a second amplifier that makes it strong enough that, when it’s fed into a solenoid, it makes the solenoid’s core move. In moving, the core strikes the hammer, the hammer loses its balance and swings downward, hitting the toggle switch and thereby turning on the light.

Everything described in that paragraph lies in the classical realm. We have a pulse of electricity moving in accordance with Ohm’s law and generating a magnetic field in the solenoid in accordance with Maxwell’s Equations. The solenoid’s core and the hammer move in accordance with Newton’s laws of motion and the hammer moves also in accordance with the law of gravity. And then we come back to the classical laws of electricity in the circuit that includes the light bulb. We only come back to the quantum realm in the understanding that the hot filament in the light bulb sheds light more or less in accordance with Max Planck’s theory of blackbody radiation.

We thus infer that the quantum-to-classical transition took place somewhere in the Geiger counter and the Geiger-Müller tube is our prime suspect. We have a quantum-scale entity, the magnesium atom, going into it and a classical-scale entity, the pulse of electric current, small though it may be, coming out of it. And between those entities we have a single electron, coming out of the magnesium atom and causing thousands of ionizations: in that electron’s behavior we see the existential probabilities of the quantum theory segue into the statistical probabilities of modern thermodynamics. Indeed, the randomness assumed in statistical dynamics likely reflects the underlying quantum indeterminacy.

Of course, we can follow the electron through its quantum description: it remains a quantum-scale object after all. But every time the electron interacts with another object, as by striking an atom and knocking off another electron, its probability fields change discontinuously in a quantum jump. There’s no way in which we can anticipate the result of a quantum jump, so there’s no way in which we can calculate out a trajectory, however fuzzy, for the electron.

At this point it will help us to look at the kind of scale on which our electron exists and moves. In the 1961 double-slit experiment which demonstrated that electrons undergo the same diffraction and self-interference that we detect in light passing through a double slit the slits each had a width of 0.3 micron (a wavelength of ultraviolet light) and were spaced one micron (a wavelength of near infrared light) apart. If we pick a point fifty microns from the electron’s center of probability, we will find the electron’s probability of existing at or near that point differs very little from zero, so we can assert that almost all of the electron’s trajectory lies inside an imaginary tube with a diameter of one hundred microns, roughly equal to the thickness of a sheet of 20-lb paper. If we could see a trace of the electron’s trajectory within the gas filling the Geiger-Müller tube, we would see the trajectory kinked up into a series of randomly-oriented, minuscule zig-zags.

But the actual trajectory that the electron follows is irrelevant to our experiment. We actually want to know that the electron will ionize atoms in the gas and enough of them to instigate a Townsend avalanche big enough for the Geiger counter to detect. We want a kind of expectation value for the number of ionizations the emitted electron will cause.

We can certainly use the quantum theory to calculate the probabilities of the electron colliding with atoms in the gas and ionizing them and use those probabilities to calculate the desired expectation value. But we can also use statistical thermodynamics, which is based on classical Newtonian dynamics, to make essentially the same calculation. Here we see the quantum theory and classical dynamics overlapping each other as one segues into the other as a proper and correct description of Reality. And we see that transition because here we have particles’ actions growing very large relative to Planck’s constant.

However that may be, we still have not answered the question that Schrödinger raised through his imaginary experiment. In our case, does the existence of a magnesium atom in a superposition of states (neither decayed nor undecayed) necessitate the other parts of the experimental apparatus existing in a superposition of states? Does the gas in the Geiger-Müller tube, for example, exist in a state of neither ionized nor un-ionized?

Does an indeterminate state associated with one object create or shape an indeterminate state in neighboring objects? Putting Schrödinger’s question into that form lets us see more clearly the logical path to an answer. Our first object was a magnesium atom when we put it into the Geiger-Müller tube: the data we gained from measurements of the interaction of placing the object are consistent with an interpretation of the object as a magnesium atom. After that interaction the magnesium atom’s probability fields begin evolving anew, in accordance with Equation 4, into the potential to be a magnesium atom and the potential to be an aluminum atom plus an electron and an anti-neutrino, both potentials standing with respect to the next interaction. We want that next interaction to be the one in which the object interacts with its own parts and commits beta decay, thereby turning a potential aluminum atom into an actual aluminum atom and erasing the potential magnesium atom.

Does that superposition of a potential magnesium atom and a potential aluminum atom transform the surrounding gas into potentially ionized and potentially un-ionized? If so, then we should expect that effect, as Schrödinger did, to extend further along the experimental apparatus until we find, let’s say, the hammer in a superposition of states being potentially upright and potentially fallen. But we can see that the hammer is actually upright and potentially fallen only in the classical sense that a slight nudge will make it fall. We can see that the hammer does not go into an indeterminate state because the light by which we see the hammer interacts more or less continuously with it, committing actions that are overwhelmingly greater than Planck’s constant. Although the probability fields of the particles that constitute the hammer evolve and collapse normally (in accordance with the quantum theory), any probability field that we might assign to the hammer itself, conceived as a collection of particles, will collapse as soon as it is formed: the hammer exists in a continuous eigenstate (upright) that is indistinguishable from the classical description.

Nothing in that analysis changes if we hide the hammer inside a box, as Schrödinger’s experiment obliges us to do. The heat radiation inside the box provides the interactions that keep the hammer in a continuous eigenstate. Putting the box into one of the great voids in intergalactic space won’t change that analysis. The Cosmic Background Radiation will heat the box enough to keep the hammer actualized.

Now let’s return to the place in our apparatus where we located the quantum-classical transition, the gas that fills the Geiger-Müller tube. In that gas the electron emitted in the decay of the magnesium atom participates in thousands of collisions, thereby instigating a Townsend avalanche. Each collision is a quantum event, but it also sounds classical: a small body strikes a larger object, bouncing off it and knocking off a small chunk at the same time. In mass action, then, the gas presents us with a quantum system that obeys the laws of statistical mechanics.

But that still doesn’t quite answer the question: Is the gas quantum or classical? I say that it’s classical, that it does not exist in an indeterminate state. Until the magnesium atom decays the gas is actually in the un-ionized eigenstate and potentially in the ionized state. It exists in that determinate state because it is, in essence, under continuous observation by the Geiger counter through the electric field that exists between the cathode and the anode in the Geiger-Müller tube. The electric field, conforming to the classical physics of Maxwell’s Equations, doesn’t produce any action in the un-ionized state, but in the ionized state, through the Townsend avalanche, creates enough action to swamp Planck’s constant.

Now we can see where the quantum-classical transition occurs. If we have a unified system large enough that its parts undergo interactions with their environment more or less continuously, then we have a system that cannot enter or remain in an indeterminate state: the system is indistinguishable from a classical object. Thus the quantum indeterminacy of the magnesium atom stops at the Geiger-Müller tube and the light is either on or off but not neither.

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