A simple, practical experiment to investigate atomic wavefunction reduction within a Stern–Gerlach magnet

For nearly one hundred years most quantum-mechanics texts have depicted a continuous, spin-direction superposition of the wavefunction of the silver atom traversing a Stern–Gerlach (S–G) magnet. But there are sound scientific arguments which deny that understanding. Schrodinger’s equation for that continuous wavefunction development cannot describe the transfer of energy from the magnetic field to that atom. Modern micro-fabrication techniques now make it possible to implement a simple, accessible experiment testing whether a spin superposition does continue in the magnet. Because the S–G experiment is the prototype for quantum measurement, that observation is crucial to implementing a realistic quantum theory of measurement.


Introduction
It has long been the accepted explanation [1][2][3] that the spin wavefunction of a neutral, spin one-half atom remains in a quantum superposition of opposite spin directions, and opposite magnetic moments, as it passes through a Stern-Gerlach (S-G) magnet.Even though the atom is successively kicked to just one side of the incident beam direction, denoted here as, ŷ, as it absorbs field quanta within the magnet, it is, nevertheless, supposed to remain in that quantum spin superposition, 1 √2 (|↑ z ⟩ + ⟨↓ z |), until striking a detector screen beyond the magnet.
That received explanation can now be tested with a simple, practical experiment.The S-G experiment [4] is a prototype of quantum measurement, and measurement is an essential ingredient of quantum theory.But there is no consensus among physicists about what, exactly, constitutes a quantum measurement.So, correcting the almost century-old belief in a continuous, spin-direction superposition throughout the S-G magnet is a seminal first step in better understanding quantum mechanics.

An experiment
A single, neutral, spin one-half atom of mass, m, and magnetic moment µ, transiting a S-G magnet with constant field B(r), experiences a time-independent potential, V (r) = µ * B (r).The Schrodinger's equation is then ). Assume a separable solution for the atomic wavefunction, Ψ (r, t) = Φ (r) Θ (t).So, Schrodinger's equation becomes, . Note that one side of the equation depends only on time, and the other side only on position.Thus, each side of the equation must equal the same constant, which is the expectation value of the total atomic energy, E. There can be continuous Schrodinger evolution of the wavefunction through the magnet, only so long as the atom's total energy, E, remains constant.But, the atom's total energy is not constant while it travels through the magnet.Each of the innumerable magnetic field quanta absorbed by the atom augments its kinetic, and total real energy, kicking the atom to one side of its initial trajectory.Schrodinger's continuous equation does not describe such quantized energy exchange [5].And, astute quantum theorists have told us so.According to von Neumann, '…the time dependent Schrodinger differential equation… describes how the system changes continuously and causally in the course of time, if its total energy is known' [6].Eugene Wigner reiterated, 'In quantum mechanics, as in classical physics, we postulate the existence of isolated systems.In both theories, if a complete description of an isolated system is given at one time, a complete description for any other time is uniquely determined as long as the system remains isolated-i.e. is not influenced by any other system.In this sense, both systems are deterministic.' [7].
Thus, wavefunction development in the Stern-Gerlach magnet is discontinuous when the atom absorbs real energy from the magnet.Discontinuity in wavefunction evolution implies a loss of interference downstream of the S-G magnet (see the appendix for details).Now, using modern micro-fabrication techniques, it is possible to determine whether such interference does exist beyond the magnet.
It is only in recent times that it is become possible to test, experimentally, whether a continuous, spin superposition, traversing the S-G magnet, is actual.It was Bohm [2], then Wigner [8], long ago, who proposed such an experiment.They supposed that if the two putative, separated wavepackets of the atom could be precisely recombined, both in position and momentum, beyond the S-G magnet, with additional bending magnets, the initial, incident, atomic wavefunction would be restored there.
If every atom's incident spin direction were prepared in x, say, then after the assumed restoration, every downstream measurement (with an additional S-G magnet, for instance), would record an x spin direction.However, if the reality is immediate spin-direction reduction in an S-G magnetic field, equally likely to be toward, then one will measure an equal number of atoms with x and −x spin instead.Some years ago, this suggested measurement was considered carefully, in detail, by Scully, Schwinger, and Englert.In a series of three papers [9][10][11], they found it would be impractical to so carefully control the downstream, bending magnets that they could precisely recombine the two, supposed wavepackets, and then actually restore the original wavefunction with its characteristic spin direction.If the wavefunction of a single silver atom remained in a spin-direction superposition within the S-G magnet, each coherent wavepacket would be diffracted by the depicted, narrow slits, and contribute to an interference pattern.But if each atom's wavefunction is immediately reduced to just one spin-direction wavepacket in the magnet, no interference would be seen.
But, now, a similar experiment is, indeed, practical, without the bending magnets.A very narrow slit in an otherwise opaque screen, capable of diffracting silver (or sodium, potassium, or rubidium) atoms traveling several hundred meters per second through an S-G magnet [12] can be micro-fabricated.We could position such a slit on each of the two beams emerging from an S-G magnet, allowing the two wavefunctions to overlap (see figure 1).This simple arrangement can be used to search for the interference pattern which would characterize the putative, coherent wavepackets of each atom, simultaneously traveling two separated paths through the magnet.But, no such interference pattern would be found if each atom's wavefunction, with its own unique quantum phase, went just one way, or the other, through the field.
Remarkably, this crucial experiment, simple and inexpensive, has yet to be implemented.It could be performed in many university laboratories that already possess a functioning S-G magnet.
We may use potassium atoms from an oven, rather than silver, as in the original S-G experiment, to perform this experiment.Potassium has a boiling point lower than silver (1032 K, rather than 2435 K), so the most probable velocity of potassium atoms vaporized in the oven, given by the Maxwell-Boltzmann distribution as v = 645 m s −1 , is slower than silver atoms through the magnet.Consequently, they will experience greater deflection.A modern S-G magnet can produce a field gradient of at least 1000 T m −1 .Thus a magnet 3.5 cm long would separate the two trajectories by about .28mm at the exit.If a detector were 50 cm from the magnet the two atomic trajectories would be about 2 mm apart there.
And, it is now possible to microfabricate slits in an opaque screen that are as narrow as 20 nm, separated by .28mm.Such a screen can be employed to distinguish a spin-direction superposition of two atomic wavepackets from wavepackets of single atoms traversing an S-G magnet along one, or the other, path.The quantum wavelength of a potassium atom moving at v = 645 m s −1 is λ = h m0v = 1.58 × 10 -11 m, where m 0 is the potassium mass, 6.49 × 10 −26 kg.If single atoms travel one or the other of the two paths through the S-G magnet to such a slit, their quantum wavelength contributes to a diffraction pattern at each of two separated locations at the downstream detector.With a slit width of 20 nm, and the detector 500 mm beyond the slit screen, the pattern resulting from many individual, vaporized potassium atoms each passing one of the slits would show two peaks of approximately 0.1 mm FWHM separated by 2 mm.
If, instead, two superposed wavepackets from each atom illuminated both slits, the Fraunhofer interference pattern far from the two slits would show a single, detected central peak of about 0.1 mm FWHM.
Note that a very narrow interference peak would appear at the center of the detector if a spin-direction superposition persisted through the magnet.But, single potassium atoms following one or the other trajectory to a slit, create an observed pattern of two peaks, each 1 mm away from the center.The difference in those two patterns is clearly observable.

The Folman experiment
Now, however, a research group has observed interference from a cloud of some ten thousand Rb atoms, prepared in a Bose-Einstein condensate (BEC), traversing an S-G field [13,14].They have built an atom trap, operating below a millionth degree absolute, which confines and controls the BEC cloud.They have also made an extraordinary atom chip to precisely channel the BEC through their tiny S-G magnet, and then accurately manipulate the two emerging beams.Those two beams travel a short distance while each atom's wavepacket expands.The wavefunctions from each beam overlap the other beam's wavefunctions at a detector.A clear, high-visibility interference pattern is recorded.
Because all the atoms from a BEC have quantum coherence, it is distinct atoms, among the ten thousand in each cloud, traveling separate paths through the S-G field, which subsequently interfere with adjacent atoms to produce the interference pattern.The researchers have not demonstrated that each Rb atom's wavefunction is split into two coherent wavepackets, analogous to individual photons passing a double-slit screen, yielding interference.
These researchers are also able to precisely recombine the wavepackets of Rb atoms from the two beams exiting the chip, in both position and momentum [15], as Wigner originally suggested, and then test for quantum coherence.But, instead of checking whether every atom has retained its original spin direction, with an additional S-G apparatus, they employ Ramsey's oscillating fields method [16], which is sometimes mistaken for an interferometer.Though the Ramsey procedure produces a sinusoidal curve of spin-flip probability, it is similar to an interference pattern in superficial appearance only [17].It is not a valid test of quantum coherence.However, if a spin superposition did actually persist through an S-G magnet, the simple experiment proposed here would confirm it.

Conclusion
Quantum measurement is a key to understanding our physical world, and the S-G experiment is the prototype of that measurement.But, it has been debilitated by the received explanation of a continuous spin superposition traversing the magnet.As usual in science, a critical observation could correct a misunderstanding.The remarkably simple, practicable experiment suggested here can do that.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Appendix. Process 1
Discontinuous Schrodinger time evolution of an atom's wavefunction means that there is an instantaneous jump in the wavefunction; the wavefunction changes discontinuously in time.Von Newmann called it 'process 1' [18].It is the simplest possible 'mechanism' of wavefunction development, the same one as the photon absorption which kicks an atom's electron orbital to an excited state.In a S-G magnet it is the atom's linear momentum which jumps discontinuously.
We are told that Schrodinger despised the idea of quantum jumps [19], but they are, nevertheless, real.Absorption of a field quantum results immediately in a single spin direction and a classical trajectory along one path or the other, of an atom through an S-G magnet.A field gradient in the S-G magnet produces a force which accelerates atoms in a direction transverse to the initial velocity, increasing their total energy.Wavefunction time development, determined by Schrodinger's equation, is discontinuous at energy exchange.The atomic wavefunction collapses there denying downstream interference.Yet, even as quantized momenta are transferred to the atom from the S-G magnetic field, the atom's path develops continuously.
A quantum wavefunction can always be written as a sum over the complete set of its energy eigenfunctions.Schrodinger time development for a static potential is discontinuous when the total energy increases, since a new, distinct wavefunction, the sum of new energy eigenfunctions, is created 2 .
Instantaneous energy transfer is substantiated by recent, careful measurements done at the National Institute of Science and Technology in the USA few years ago, David Weinland, the Nobel laureate, and his colleagues, reported macroscopic quantum jumps seen as intermittent fluorescence of trapped atomic ions [20].The time required for the energy transition, they found, is smaller than can be measured with the best instruments available.Those experimentalists emphasized that the process is as nearly instantaneous as can be determined today; not a continuous one.If a quantum of energy, such as a photon, could be transferred as contiguous, partial pieces over some tiny time interval, it would not be a discrete quantum.
The classical trajectory that an atom follows through the S-G magnet can be computed.The time evolution operator for an infinitesimal time, δt, between each discontinuous momentum jump is Ψ (r, t + δt) = [1 − i ℏ Hδt]Ψ (r, t).A realistic magnetic field that satisfies Maxwell's equations, B(r) = (−bx, 0, bz + B 0 ), with b and B 0 constant, is employed in the Hamiltonian, H.A minimum uncertainty wavefunction is assumed initially.With the atom in the beam line at t = 0, we write the wavefunction in two dimensions as ), with w x and w z the initial, corresponding widths.The atomic wavepacket exiting the magnet can be computed from these initial conditions.The time required for transit of the field is divided into many infinitesimal intervals, δt , and the time evolution process is iterated.A detailed computer computation by Hsu et al [21] shows a pattern on the detector that conforms to that first seen by Stern and Gerlach.But these researchers have presumed two coherent wavepackets of opposite spin directions transiting the magnet simultaneously.That same computation, recognizing, instead, just a 2 The quantum measurement process within a S-G magnet determines the atom's spin-direction eigenfunction and a new wavefunction, with a unique quantum phase.Other, somewhat similar phenomena, when no energy is exchanged, do not reduce a wavefunction to a single eigenfunction.There is, for instance, no energy transferred from a single photon to the double slits which divide that wavefunction into two superposed wavepackets.Likewise, the beam splitter of an interferometer, or a birefringent crystal, continuously divides the wavefunction into separated wavepackets, with no abrupt wavefunction reduction, or measurement.Even a parametric, downconverting crystal [22], ubiquitous in quantum-information observations, preserves the total energy of the incident photon.
single spin-direction wavepacket each time, also produces that same pattern for discontinuous wavefunction development in the S-G experiment.

Figure 1 .
Figure 1.If the wavefunction of a single silver atom remained in a spin-direction superposition within the S-G magnet, each coherent wavepacket would be diffracted by the depicted, narrow slits, and contribute to an interference pattern.But if each atom's wavefunction is immediately reduced to just one spin-direction wavepacket in the magnet, no interference would be seen.