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Quantum mechanical modeling of the multi-stage
Stern–Gerlach experiment conducted by Frisch
and Segrè
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et al
2024
New J. Phys.
26 073005
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New J. Phys.
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PAPER
Quantum mechanical modeling of the multi-stage Stern–Gerlach
experiment conducted by Frisch and Segrè
S Süleyman Kahraman
1
,
3
, Kelvin Titimbo
1
,
3
, Zhe He
1
, Jung-Tsung Shen
2
and Lihong V Wang
1
,
∗
1
Caltech Optical Imaging Laboratory, Andrew and Peggy Cherng Department of Medical Engineering, Department of Electrical
Engineering, California Institute of Technology, 1200 E. California Blvd., MC 138-78, Pasadena, CA 91125, United States of America
2
DepartmentofElectricalandSystemsEngineering,WashingtonUniversityinSt.Louis,St.Louis,MO63130,UnitedStatesofAmerica
3
These authors contributed equally.
∗
Author to whom any correspondence should be addressed.
E-mail:
lvw@caltech.edu
Keywords:
spin-flip transitions
,
electron spin
,
quantum dynamics
,
nonadiabatic transitions
,
hyperfine interaction.
Abstract
The multi-stage Stern–Gerlach experiment conducted by Frisch and Segrè includes two cascaded
quantum measurements with a nonadiabatic flipper in between. The Frisch and Segrè experiment
has been modeled analytically by Majorana without the nuclear effect and subsequently revised by
Rabi with the hyperfine interaction. However, the theoretical predictions do not match the
experimentalobservationaccurately.Here,wenumericallysolvethestandardquantummechanical
model, via the von Neumann equation, including the hyperfine interaction for the time evolution
of the spin. Thus far, the coefficients of determination from the standard quantum mechanical
model without using free parameters are still low, indicating a mismatch between the theory and
the experiment. Non-standard variants that improve the match are explored for discussion.
1.Introduction
The quantum measurement problem tackles the conundrum of wave function collapse and the
Stern–Gerlach (SG) experiment is considered as the first observation of a quantum measurement [
1
–
6
].
While the SG observation was interpreted as proof of quantization of spin [
7
–
9
], cascaded quantum
measurements provide a more stringent test of theories [
3
,
10
]. Frisch and Segrè (FS) conducted the first
successful multi-stage SG experiment [
1
,
11
–
13
] after improving the apparatus from Phipps and Stern [
14
].
Even though more recent multi-stage SG experiments have been conducted, they differ in the mechanisms of
polarizing, flipping, and analyzing the spin [
15
–
22
]. Most experiments designed for precise atomic
measurements use a narrow-band resonant flipper [
15
] while the FS experiment uses a wide-band
nonadiabatic flipper.
The FS experiment was suggested by Einstein [
8
,
13
,
23
], studied analytically by Majorana [
24
,
25
] and
later by Rabi [
26
]. Majorana investigated the nonadiabatic transition of the electron spin through a
closed-form analytical solution, which is now widely used to analyze any two-level systems [
27
]. Rabi revised
Majorana’s derivation by adding the hyperfine interaction but still could not predict the experimental
observation accurately. Despite additional theoretical studies into similar problems involving multilevel
nonadiabatic transitions [
27
–
33
], an exact solution with the hyperfine interaction included has not been
obtained.
Among the more recent multi-stage SG experiments [
16
–
21
], the study most similar to the FS
experiment uses a sequence of coils to obtain the desired magnetic field [
16
,
17
]. The models in these works
not only simplified the mathematical description of the magnetic fields generated by the coils but also fit free
parameters to predict the experimental observations. We choose to model the FS experiment over other
similar experiments because of the simplicity of the nonadiabatic spin flipper and its historical significance.
Here, we numerically simulate the FS experiment using a standard quantum mechanical model via the
von Neumann equation without tuning any parameters and compare the outcome with the predictions by
both Majorana and Rabi. Even though our approach is a standard method of studying such spin systems, our
©2024TheAuthor(s). PublishedbyIOPPublishingLtdonbehalfoftheInstituteofPhysicsandDeutschePhysikalischeGesellschaft
New J. Phys.
26
(2024) 073005
S S Kahraman
et al
resultsdonotmatchtheexperimentalobservations.Thisdiscrepancyindicatesthateitherourunderstanding
of the FS experiment is lacking or the standard theoretical model is insufficient. Recent studies have modeled
the FS experiment using an alternative model called co-quantum mechanics [
34
–
36
] without resorting to
free parameters. Despite the existence of several theories regarding the reduction of the wavefunction that
couldpotentiallyaddresstheobservedmismatch[
37
–
41
],thisstudyremainswithintheorthodoxframework
of the standard quantum theory and the Born postulate [
3
–
6
].
This paper is organized as follows. In section
2
, we present the experimental configuration used by Frisch
and Segrè to measure the fraction of electron spin flip. In section
3
, we introduce the von Neumann equation
and the Hamiltonian for the nuclear-electron spin system. Numerical results for the time evolution of the
spins and the final electron spin-flip probability are shown here. In section
5
, we compare the numerical
results with previous solutions. Finally, section
6
is left for conclusions. Non-standard variants of the
quantum mechanical model are explored in the appendices to stimulate discussion.
2.DescriptionoftheFrisch–Segrèexperiment
The schematic of the setup used in the Frisch–Segrè experiment [
11
,
12
] is redrawn in figure
1
. There,
magnetic regions 1 and 2 act as Stern–Gerlach apparatuses, SG1 and SG2, respectively, with strong magnetic
fields along the
+
z
direction. In SG1, stable neutral potassium atoms (
39
K) effused from the oven are
spatially separated by the magnetic field gradient according to the orientation of their electron magnetic
moment
μ
e
. The magnetically shielded space containing a current-carrying wire forms the inner rotation
(IR) chamber. The shielding reduces the fringe fields from the SG magnets down to the homogeneous
remnant field
B
r
=
42
μ
T aligned with
+
z
. Inside the IR chamber, a wire placed at a vertical distance
z
a
=
105
μ
m below the atomic beam path carries a time-independent current, creating a cylindrically
symmetric magnetic field. The total magnetic field in the IR chamber equals the superposition of the
remnant field and the magnetic field created by the electric current
I
w
flowing through the wire. The precise
magnetic field outside the IR chamber was not reported [
11
,
12
]. After SG1, the atoms enter the IR chamber;
we approximate the motion as rectilinear and uniform along the
y
axis. The rectilinear approximation of
atomic motion within the IR chamber is acceptable since the total displacement due to the field gradients is
negligible, approximately
1
μ
m. In the FS experiment, the magnetic field is time-independent. However, by
approximating the atom as in an inertial reference frame, we consider the field experienced by the atom to be
time-dependent. Along the beam path, the magnetic field is given by
B
exact
=
μ
0
I
w
z
a
2
π
(
y
2
+
z
2
a
)
e
y
+
B
r
−
μ
0
I
w
y
2
π
(
y
2
+
z
2
a
)
e
z
,
(1)
where
μ
0
is the vacuum permeability; the trajectory of the atom is expressed as
y
=
vt
, where
v
is the speed of
the atom and the time is set to
t
=
0 at the point on the beam path closest to the wire. The right-handed and
unitary vectors
e
x
,
e
y
,
e
z
describe the directions of the Cartesian system depicted in figure
1
.
The magnetic field inside the IR chamber has a current-dependent null point below the beam path at
coordinates
(
0
,
y
NP
,
−
z
a
)
, with
y
NP
=
μ
0
I
w
/
2
π
B
r
. In the vicinity of the null point, the magnetic field
components are approximately linear functions of the Cartesian coordinates. Hence, the magnetic field can
beapproximatedasaquadrupolemagneticfieldaroundthenullpoint[
11
,
24
].Alongtheatomicbeampath,
the approximate quadrupole magnetic field is [
34
,
36
]
B
q
=
2
π
B
2
r
μ
0
I
w
z
a
e
y
+
2
π
B
2
r
μ
0
I
w
(
y
−
y
NP
)
e
z
.
(2)
For the study of the time evolution of the atom inside the IR chamber both of the fields,
B
exact
and
B
q
, are
considered below.
After the IR chamber, a slit transmits one branch of electron spins initially polarized by SG1 and blocks
the other branch. The slit was positioned after the intermediate stage to obtain a sharper cut-off [
11
]. In the
forthcomingtheoreticalmodel,wetrackonlythetoptransmittedbranch,correspondingtothoseatomswith
spin down (
m
S
=
−
1
/
2
), at the entrance of the IR chamber and ignore the branch blocked at the exit.
However, the opposite choice of branch (
m
S
=+
1
/
2
) yields exactly the same results in this model. The atoms
that reach SG2 are further spatially split into two branches corresponding to the electron spin state with
respect to the magnetic field direction. The final distribution of atoms is measured by scanning a hot wire
along the
z
axis while monitored by the microscope. The probability of flip is then measured at different
values of the electric current
I
w
.
2
New J. Phys.
26
(2024) 073005
S S Kahraman
et al
Figure1.
Redrawn schematic of the original setup [
11
,
12
]. Heated atoms in the oven effuse from a slit. First, the atoms enter
magnetic region 1, which acts as SG1. Then, the atoms enter the region with magnetic shielding (i.e. the inner rotation chamber)
containing a current-carrying wire W. Next, a slit selects one branch. Magnetic region 2 acts as SG2. The hot wire is scanned
vertically to map the strength of the atomic beam along the
z
axis. The microscope reads the position of the hot wire.
Figure2.
Schematicofthemodelconsideredinthisstudy.ThesystemconsistsoftwomeasurementswithSG1andSG2.Theinner
rotation chamber that induces nonadiabatic transitions is modeled with the von Neumann equation. The evolution from the end
of SG1 and the filter to the entrance of the rotation chamber is modeled as an adiabatic evolution. Similarly, the evolution from
the exit of the inner rotation chamber to the entrance of SG2 is modeled as adiabatic evolution.
3.Theoreticaldescription
The time evolution of the noninteracting atoms in the beam traveling through the IR chamber of the
Frisch–Segrè experiment is studied using standard quantum mechanics. The whole setup is modeled in
multiplestagesasillustratedinfigure
2
.First,theoutputofSG1andtheslitismodeledasapureeigenstateof
the electron spin measurement in the
z
direction. Since the gradient of the strong field in SG1 is not high
enough, nuclear spin eigenstates do not separate during the flight. Hence, the nuclear state is assumed to be
unaffected by SG1 and the slit. During the flight from SG1 to the entrance of the IR chamber, the state is
assumed to vary adiabatically as in figure
2
. The magnetic fields in the transition regions were not reported;
but when the IR chamber was turned off,
I
w
=
0A, no flipping was observed after SG2 [
12
]. Therefore, it can
beassumedthatoutsidetheIRchamber,thestateevolvesadiabatically.Later,theatomenterstheIRchamber
designed to induce nonadiabatic transitions. The evolution of the state in the IR chamber is modeled using
the von Neumann equation, which is solved using numerical methods. During the flight from the exit of the
IR chamber to SG2, the state is assumed to vary adiabatically as in figure
2
. Finally, SG2 measures the
probabilities in different electron spin eigenstates in the
z
direction according to the Born principle.
The density operator formalism is used for its capability to represent mixed states in quantum systems,
offering a more complete description than pure states alone [
42
,
43
]. The time evolution of the density
operator
ˆ
ρ
is governed by the von Neumann equation:
∂
ˆ
ρ
(
t
)
∂
t
=
1
i
̄
h
[
ˆ
H
(
t
)
,
ˆ
ρ
(
t
)]
,
(3)
where
ˆ
H
(
t
)
is the Hamiltonian of the system and
̄
h
is the reduced Planck constant. For the time-dependent
Hamiltonian
ˆ
H
(
t
)
, we introduce the instantaneous eigenstates
ψ
j
(
t
)
and eigenenergies
E
j
(
t
)
such that
ˆ
H
(
t
)
ψ
j
(
t
)
=
E
j
(
t
)
ψ
j
(
t
)
,
(4)
3
New J. Phys.
26
(2024) 073005
S S Kahraman
et al
where
j
can take a finite number of values for the spin system considered here. In the basis of the
instantaneous eigenstates of the Hamiltonian, from (
3
) the matrix elements of the density operator,
ρ
j
,
k
(
t
)=
ψ
j
(
t
)
|
ˆ
ρ
(
t
)
|
ψ
k
(
t
)
, evolve according to
∂ρ
j
,
k
(
t
)
∂
t
=
"
1
i
̄
h