FOUCAULT’S PENDULUM, A CLASSICAL ANALOGFOR THE ELECTRON SPIN STATE by Rebecca A.. FOUCAULT’S PENDULUM, A CLASSICAL ANALOGFOR THE ELECTRON SPIN STATE by Rebecca A.. To bridge the gap and
Foucault and His Pendulum
Foucault’s pendulum is named for French inventor Jean Bernard L´eon
Foucault (1819–1868) is described by Baker and Blackburn as having initially studied medicine but spending most of his life in the physical sciences His interest in optics led him to study interference, chromatic polarization, and the speed of light through different media To support these investigations, he invented and improved several devices, notably a precision gyroscope and techniques that enhanced telescope manufacture.
In 1851, while working on a telescope, Foucault made a surprising discovery: the plane of oscillation of a vibrating rod fixed in a lathe chuck remained fixed in orientation even as he slowly turned the chuck Baker and Blackburn summarize this observation, showing how a simple experimental setup can reveal fundamental aspects of motion, and noting that the result spurred follow-up experiments that ultimately demonstrated the Earth's rotation.
Foucault to wonder if the orientation of the plane of a simple pendulum’s oscillation might similarly be affected by the earth’s rotation.
In January 1851, Foucault tested his idea with a 2-meter Foucault pendulum in his mother's basement As the 5-kilogram bob swung, the plane of oscillation slowly precessed, providing a tangible demonstration of the Earth's rotation He then sought to improve the experiment with larger pendulums, culminating in a 68-meter-long pendulum with a 28-kilogram bob suspended from the dome of the Pantheon in Paris, whose demonstration convinced the general public of the Earth's rotation.
Today, Foucault pendulums are on display in museums and science centers around the world Although any planar pendulum laid out as in the diagram can exhibit Foucault-like motion, demonstrations typically use very massive bobs and long arms to maximize the effect To keep the motion going despite friction and air resistance, many exhibits incorporate a device that periodically gives the pendulum a gentle push Because the rate of precession is constant—set by the pendulum’s latitude and the Earth’s rotation—pegs are often arranged along the edge of the pendulum’s circular path As the pendulum swings, its precession causes it to knock over the pegs, helping observers notice the slow rotation.
The Classical Lagrangian
We begin with the Lagrangian that describes the motion of Foucault’s pendulum, using it as the starting point to analyze the system’s dynamics Because the pendulum is observed in a non-inertial reference frame, its Lagrangian is more complicated than the standard single-particle Lagrangian Following Landau and Lifshitz, the Lagrangian for a particle in a non-inertial reference frame can be expressed in a form that accounts for the effects of the non-inertial frame on the dynamics.
In a non-inertial reference frame, the particle’s Lagrangian differs from the standard L = 1/2 m v^2 − U by three additional terms If r and v denote the particle’s position and velocity, and Ω is the frame’s angular velocity while W is its translational acceleration, then the non-inertial Lagrangian can be written as L' = 1/2 m v^2 − U plus those extra terms that arise from the frame’s motion Comparing L' with the conventional single-particle Lagrangian reveals that observing the particle in a non-inertial frame adds three new terms to the Lagrangian Among these, two have special significance: the second term corresponds to the energy imparted to the particle by the Coriolis force, and the third term corresponds to the energy imparted by the centrifugal force.
To arrive at the Lagrangian that describes the motion of Foucault’s pendulum, we begin by discarding the parts of the generalized Lagrangian (2.1) that do not pertain to the pendulum By focusing on measurements of the pendulum bob’s position, we constrain the analysis to the pendulum’s relevant degrees of freedom, yielding a reduced Lagrangian that captures the essential dynamics of the pendulum in a rotating frame.
Figure 2.2 shows a coordinate system fixed to the Earth's surface, rotating with the planet and remaining at a constant distance from the Earth's center By neglecting the Earth's orbital motion, this rotating frame experiences no translational acceleration, so W = 0 Since the particle's potential energy arises from gravity, U = - m g · r, the Lagrangian can be reformulated under these conditions.
To capture the pendulum’s dynamics, we express the bob’s position and velocity in Cartesian coordinates, as shown in Fig 2.2 The Earth's rotation frequency vector Ω is then decomposed into components parallel and perpendicular to the plane of the Earth’s surface Substituting these expressions into equation (2.2) yields the Lagrangian of the system.
2m( ˙x 2 + ˙y 2 + ˙z 2 ) +mΩ [( ˙yx−xy) cos˙ α+ ( ˙yz−zy) sin˙ α]
Figure 2.3: The Position of the Pendulum Bob Relative to the Earth’s Surface
Now, we use Fig 2.3 to re-express the pendulum bob’s vertical motion in terms of the length of the pendulum arm, z = 2` 1 (x 2 +y 2 +z 2 )
In the small angle limit, the motion of the pendulum is restricted to the xy-plane, (so z ≈0 and ˙ z ≈0) Using this approximation and defining a new variable β = Ω cosα, the Lagrangian becomes
` is the oscillation frequency of a simple planar pendulum.
Earth’s rotation is relatively slow, with an angular velocity of 2π radians per day (approximately 7.3×10^-5 rad/s), so terms that scale with Ω^2 are typically dropped from the expansion By neglecting these Ω^2 terms, the Lagrangian simplifies to its leading-order form, removing the quadratic-in-rotation contributions and leaving a reduced expression that captures the essential dynamics at this order.
This Lagrangian encodes the dynamics of Foucault’s pendulum as a pair of coupled harmonic oscillators, with x and y representing the coordinates of the two oscillators and β acting as the coupling parameter between them When β is set to zero, the coupling vanishes and the Lagrangian reduces to that of two independent, uncoupled harmonic oscillators, as described in equation (2.5).
Electron Spin State in a Constant 1D Magnetic Field
Now, performing a change of variables, let x be x1 and y be x2 Doing this, the Lagrangian in equation (2.5) becomes:
, it is possible to arrive at the following pair of coupled equations of motion: ¨ x 1 = 2βx˙ 2 −w 2 0 x 1 ¨ x 2 =−2βx˙ 1 −w 2 0 x 2 (3.2)
These equations, once solved, yield the real part of
Where a and b are arbitrary complex quantities andω± =p ω 2 0 +β 2 ±β.
Placing an electron in a magnetic field causes its spin state |χ_i⟩ to interact with the external field, and this interaction is described by the time-dependent Schrödinger equation Following Griffiths, the evolution is governed by iħ ∂t |χ_i⟩ = H |χ_i⟩, where H is the Hamiltonian that encodes the coupling between the spin and the magnetic field, determining how the spinor |χ_i⟩ changes over time in the magnetic environment.
Typically, the interaction is described by the Hamiltonian H = −(1/2) γ B · σ, where γ = − e / m_e is the electron’s gyromagnetic ratio, B is the magnetic-field vector, and σ is the vector of Pauli spin matrices.
It is a standard practice in quantum mechanics to disregard global phase. Suppose instead that the global phase of the electron’s spin state is included.
Accounting for global phase due to rest mass oscillation at the Compton frequency
~ ), the system’s Hamiltonian becomes: H=~ ω 0 I− 1 2 γBãσ
Re-expressing the magnetic field vector so that β=− 1 2 γB, the Schr¨odinger
Equation for this system becomes: i~∂
∂t|χi=~(ω 0 I+βãσ)|χi (3.4) Now, suppose that the electron is placed in a constant y-directed magnetic field The solution to the Schr¨odinger Equation for this system can be expressed as:
Written this way, a and b are complex constants and χ + and χ− are the complex parts of the spin state|χi Note that the system’s two eigenfrequencies
(ω±=ω 0 ±β) correspond to a Zeeman energy splitting of equal amount above and below the ground state.
Comparing the solution for Foucault’s pendulum given by equation (3.3) with the Schrödinger equation in (3.5) shows that the two solutions are nearly identical If the difference in their associated eigenfrequencies could be resolved, the forms would coincide Since ω0 for the pendulum is typically much larger than β, the pendulum’s normal modes are essentially ω± ≈ ω0 ± β In this limit, the standard Lagrangian for Foucault’s pendulum (3.1) maps onto the dynamics of an unmeasured spin state in a constant one-dimensional magnetic field.
An Alternate Lagrangian
Although the Lagrangian in equation (3.1) maps the spin state reasonably well in the relevant parameter regime, it would be preferable to obtain a Lagrangian that yields the correct eigenfrequencies ω± = ω0 ± β without imposing any constraint on the relative strengths of ω0 and β In other words, a more general Lagrangian formulation that inherently produces ω+ = ω0 + β and ω− = ω0 − β for all values of ω0 and β would provide a consistent description of the spin dynamics This generalization would align the Lagrangian with the expected spectrum and remove regime-dependent limitations.
Figure 3.1: Components of the Earth’s Rotation Frequency Vector with Respect to the Surface of the Earth
To do this, we return to the more general expression for the Foucault pendulum Lagrangian, as given in equation (2.4) By performing a change of variables—x becomes x1 and y becomes x2—the Lagrangian is rewritten in the new coordinates, yielding the transformed Lagrangian in terms of x1 and x2.
Equation (3.6) features a last term that depends on x2 but not on x1 Since x1 points to changes in latitude and x2 to changes in longitude, this term increases the coupling between the pendulum’s longitude and β, while there is no corresponding enhancement for latitude and β Consequently, equation (3.6) lacks symmetry The extra term, arising from the centrifugal force, slightly modifies the pendulum’s oscillation.
Retaining the final term in equation (3.6) introduces a symmetry break This symmetry breaking becomes problematic when β is interpreted as the external magnetic field acting on an electron’s spin state.
To resolve this problem, suppose instead that the x 1 and x 2 directions represent the positions of two coupled simple harmonic oscillators In this case, the appropriate Lagrangian would shows no greater dependence on the motion of particlex 1 than it would show for particle x 2 Since the last term in equation (3.6) breaks the required symmetry it is not allowed As a result, the Lagrangian that describes Foucault’s pendulum when it is represented as a pair of coupled harmonic oscillators is:
Equation (3.7) yields a Lagrangian that closely mirrors the one in equation (3.1); both describe the motion of two coupled harmonic oscillators The distinguishing addition is the term (1/2) m β^2 (x1^2 + x2^2), which uniformly amplifies the influence of the coupling parameter β across both coordinates.
Repeating the analysis performed for the Lagrangian in equation (3.1) on the Lagrangian in equation (3.7) yields the same outcome: a pair of coupled equations of motion, x¨1 = 2β x˙2 + (β^2 − ω0^2) x1 and x¨2 = −2β x˙1 + (β^2 − ω0^2) x2 (3.8).
These equations, once solved yield the real part of
This solution has a form identical to the solution of the Schrödinger equation shown in (3.5) Unlike the solution corresponding to equation (3.3), the eigenfrequencies of this solution have exactly the same form as those for the Schrödinger equation, namely ω± = ω0± β.
From the link between the pendulum solution and the spin-state solution, two conclusions follow First, retaining the β^2 term in equation (3.7)—the centrifugal-force contribution that was omitted in equation (3.1)—strengthens the correspondence between the pendulum dynamics and the spin-state map Second, the identical form of the two solutions implies that the Lagrangian in equation (3.7) describes the dynamics of an unmeasured electron spin state in a one-dimensional constant magnetic field.
Electron Spin State in a Time Dependent 1D Magnetic Field
Building on the previous success, the next phase aims to probe the limits of the correspondence between the pendulum Lagrangian and the electron spin state We advance in small steps by checking whether the Lagrangian in equation (3.7) yields the correct behavior when the external magnetic field changes over time Since β encodes the magnetic-field strength for the spin state, testing a time-varying field requires replacing β with β(t) By tracking how the system responds to β(t), we assess the fidelity of the Lagrangian-spin mapping and identify any deviations that arise as the field becomes dynamic.
Before performing this analysis, we re-express the Lagrangian from equation (3.7) in a more compact form by following the method of Wharton, Linck, and Salazar-Lazaro (WLS11), and we begin by defining a new quantity that encapsulates the relevant combinations of fields and parameters to streamline the subsequent derivation.
In this formulation, p1 and p2 represent the pendulum’s conjugate momenta The conjugate momentum can be expressed as a vector p ≡ ẋ + Bx, where x is the position vector and ẋ its time derivative (velocity); B is a matrix that couples the position and velocity terms, highlighting how the momentum depends on both x and ẋ.
B is the β matrix in equation (3.10) Using these definitions the Lagrangian in equation (3.7) can be expressed as:
Now, with this new expression for the pendulum Lagrangian, we return to the correspondence test for a time-varying magnetic field As before, we begin by applying Lagrange’s equation to the Lagrangian in (3.11) to obtain a pair of coupled equations of motion; when β is time-dependent, the resulting equations of motion are: ẍ1 − (β^2 − ω0^2) x1 = 2β ẋ2 + β̇ x2, ẍ2 − (β^2 − ω0^2) x2 = −2β ẋ1 − β ẋ1 (3.12).
Returning to the Schr¨odinger Equation in (3.4), let the spin state vector |χi be expressed in terms of the complex parameters a(t) and b(t) such that
For a time dependent y-directed magnetic field, the Schr¨odinger Equation can then be broken into these two complex equations: ˙ a+iω0a=−βb b˙+iω 0 b=βa (3.13)
Although the pendulum equations of motion (3.12) initially seem inconsistent with the spin state equations (3.13), a mapping between these two sets of equations can exist When such a map is established, the Lagrangian in equation (3.11) can describe the dynamics of an unmeasured electron spin state in a time-varying magnetic field directed along the y-axis.
We begin the map-finding process by taking the time derivative of each equation (3.13) After performing substitutions using equations (3.13), it is possible to arrive at the following set of coupled second order equations: a¨ − (β^2 − ω0^2) a = −2β ḃ − β ḃ; b¨ − (β^2 − ω0^2) b = 2β ȧ + β̇ a (3.14).
Comparing equations (3.14) to equations (3.12), it initially appears that the two sets of equations have nearly the same form However, equations (3.12) are purely real while equations (3.14) are complex.
To resolve the issue that equations (3.14) are complex, we express every complex quantity in terms of its real and imaginary parts Writing a as a = a_R + i a_I, where a_R is the real part and a_I the imaginary part, allows us to decompose (3.14) into four real second‑order equations When these real equations are combined, they yield the following pair of real equations for the spin state.
Comparing equations (3.15) with equations (3.12), it is evident that the two sets of equations are equivalent when x 1 =Re(a−ib) x 2 =Im(a−ib) (3.16)
As a result, equations (3.16) represent a map between the pendulum equations of motion (3.12) and the spin state equations (3.13).
Comparing the pendulum equations (3.12) with the spin-state equations (3.13) reveals an apparent issue with the scope of the map (3.16): the pendulum dynamics depend on two state variables (x1 and x2), while the spin-state dynamics depend on four real parameters (aR, aI, bR, and bI) At first glance this suggests that solving the spin-state system would require twice as many parameters as solving the pendulum system However, this conclusion overlooks a key aspect of second-order differential equations: solving such equations requires both an initial position (x0) and an initial velocity (ẋ0) Consequently, solving the two pendulum equations in (3.12) actually consumes four parameters, aligning with the total number of parameters needed for the spin-state description.
Inside equations (3.12) and (3.13), both the pendulum dynamics and the spin-state dynamics possess four real parameters when the state’s amplitude and global phase are retained; if those quantities are disregarded, the Schrödinger equation (3.13) generally requires only two parameters Thus the pendulum equations (3.12) and the spin-state equations (3.13) have four real degrees of freedom, and under the map (3.16) the pendulum solutions correspond to the real parts of the spin-state solutions, linking the classical and quantum descriptions Consequently, the Lagrangian (3.11) accurately describes the dynamics of an unmeasured electron spin state in a time-varying y-directed magnetic field.
Conceptual Analog
Thus far, we have restricted our discussion of the connection between
Foucault’s pendulum and the electron spin state can be compared through a mathematical description of their dynamics Before advancing the formal comparison, it is useful to examine the conceptual connections between these two systems As noted earlier, the electron spin state is widely regarded as fundamentally non-classical Yet, beyond acting as a mathematical analogy, Foucault’s pendulum provides a conceptual framework for describing many of the spin state's non-classical characteristics.
Figure 3.2: Bloch Sphere for the Electron Spin State
The Bloch sphere, also known as the Poincaré sphere in optics, provides a graphical way to describe a two-level quantum system such as the electron spin state A two-level system is defined by an orthonormal pair of basis states, and these states are mapped to opposite points on the unit sphere, with the rest of the sphere arranged so that any pair of antipodal points represents a different orthonormal basis that can describe the system Through complex superposition, any state on the sphere can be expressed in terms of any chosen pair of opposite basis states For the electron spin state, the spin-up and spin-down basis states form the standard orthonormal set and are mapped to the points where the sphere intersects the z-axis (as shown in Fig 3.2); using the Bloch-sphere properties, any spin state can thus be described as a superposition of these two basis states.
Figure 3.3: Bloch Sphere for Foucault’s Pendulum
Beyond describing the electron spin state, the Bloch sphere also represents the pendulum’s oscillation states (see Fig 3.3) The pendulum’s planar motion has two perpendicular directions—toward/away from the observer (l) and left/right perpendicular to the observer (↔)—which form an orthonormal basis and map onto the sphere’s z-axis for easy comparison As with the spin state, every other point on the Bloch sphere corresponds to a superposition of the two opposing basis states, capturing both amplitude and phase information For example, counter-clockwise rotation can be expressed as a specific superposition of the l and ↔ components, defined by a particular relative phase between them.
Expressed this way, complex numbers encode the relative phase between two orthonormal basis states of a pendulum Complex numbers are used throughout classical physics for similar purposes; for example, in circuit theory they encode the amplitude and relative phase of voltages and currents in AC circuits Since the pendulum’s classical state can be described with complex numbers, describing spin states with complex numbers does not necessarily imply that those states are inherently non-classical.
Consider a Foucault pendulum observed at a fixed instant: the observer, looking straight ahead, sees the pendulum oscillating back and forth along his sight line, placing the pendulum in the state (l) When the observer returns after some time, the pendulum remains in the same (l) state, even though the plane of its oscillation has slowly precessed during the interval Without any extra information, there is no way to tell whether the pendulum's orientation has changed by a π rotation, a 2π rotation, or some other integer multiple of π.
Suppose an observer uses the Bloch sphere in Figure 3.3 to infer the pendulum’s state at the moment of the second observation; starting from the sphere’s top and moving clockwise, the observer identifies a sequence of states that, in aggregate, completes a full rotation on the Bloch sphere This progression of states demonstrates how the pendulum’s quantum state evolves through successive measurements, with the Bloch-sphere geometry encoding the rotation and phase relationships that govern the system’s dynamics.
During the progression from l to ↔ to ↔ to l, the pendulum undergoes a full 2π rotation on the Bloch sphere, but in physical space it completes only a π rotation, leaving the pendulum out of phase with its original state and requiring an extra 2π rotation on the Bloch sphere to recover the true initial condition This discrepancy shows that the Bloch-sphere representation does not map directly onto physical space, so Figure 3.3 cannot reliably reveal the pendulum’s true state at the second observation The failure arises from a common practice in quantum theory that overlooks the amplitude and, in particular, the global phase of a quantum state—a bias embedded in the Bloch-sphere framework—meaning there is no accurate way to determine the global phase of a state once it is mapped onto the Bloch sphere.
Within quantum mechanics, it is well established that an electron spin state returns to its original orientation only after a 4π rotation This hallmark has been verified by comparing rotated and unrotated spin states If one electron's spin is rotated by 2π while another's spin is held fixed, their interaction exhibits destructive interference, indicating that the two spin states are out of phase To restore the rotated spin state to alignment with the unrotated state, an additional 2π rotation is required, bringing the total to 4π.
Revisiting the spin-state Bloch sphere in Fig 3.2 shows that the geometry would have the state return to its original orientation after a 2π rotation, yet the physical spin state actually requires a 4π rotation A similar shortcoming appears when comparing with the pendulum: in both cases the Bloch representation implies a 4π rotation to effect what should be a 2π rotation, because there is no mechanism on the sphere to track the global phase of the state With this in mind, it seems plausible that both the pendulum Bloch sphere and the spin-state Bloch sphere do not correspond to physical space.
The apparent doubling of the spin-state rotation may simply arise from neglecting the state's global phase and from interpreting Bloch-sphere rotations as physical rotations When the global phase is ignored, a twofold rotation of the spin state need not reflect a real, observable effect but rather a consequence of how the state is represented The Bloch-sphere formalism encodes rotations up to a global phase, so equating these mathematical rotations with physical motion can mislead one into seeing a doubled rotation Recognizing the role of the global phase resolves the apparent discrepancy and aligns the spin-state evolution with the correct quantum description.
According to quantum mechanics, an electron placed in a magnetic field undergoes spin precession at a frequency ω = γB, where γ is the electron’s gyromagnetic ratio and B is the external magnetic field strength With B taken as constant, this relationship directly links the gyromagnetic ratio to the observed precession frequency of the spin state If the measured precession frequency were found to be double its true value, the only parameter that can account for this change is γ, so the inferred gyromagnetic ratio would likewise be doubled.
Within the Bloch sphere framework, the standard interpretation requires that a spin state undergo a double rotation, which in turn implies that the spin’s rotation frequency is doubled This matches the fact that the electron’s gyromagnetic ratio is twice the classical value A possible explanation may lie in the pendulum Bloch sphere analysis, which could illuminate the origin of this discrepancy The doubling of the gyromagnetic ratio arises because information about the spin state's global phase has been disregarded, leading to the conclusion that the rotation frequency is twice its actual value.
Electron Spin State in a Constant 3D Magnetic Field
The Lagrangian in equation (3.11), capable of describing Foucault’s pendulum dynamics, can also model the dynamics of an unmeasured electron spin state in a time-varying one-dimensional magnetic field; however, its applicability is limited to spin–field interactions in one dimension, and handling more intricate magnetic field arrangements would require a new Lagrangian As in the development of equation (3.11), a restriction was imposed on the Lagrangian, so that it represents Foucault’s pendulum when the pendulum is modeled as two coupled harmonic oscillators If the number of oscillators is doubled while preserving the basic form, a new definition of the conjugate momentum vector p is required Suppose p is defined as
As before, p can be expressed in a more compact form as p ≡ ẋ + Bx By comparing this result with the result following equation (3.10), it is apparent that the form of p has been maintained while the underlying definitions of p, x, ẋ, and B have changed Using this definition of p, a new four-oscillator Lagrangian can be defined as:
Assuming a constant magnetic field, Lagrange's equations can be applied to derive a set of four coupled equations of motion When these equations are written in vector notation, they take the form of four interrelated vector equations that describe the system's dynamics under a fixed magnetic field.
Where, β ≡p β x 2 +β y 2 +β z 2 and B is the 4×4 matrix of β components found in equation (4.1).
By extending the method used for the two-oscillator Lagrangian (3.11), it is possible to obtain a complete set of solutions for this system We begin by rewriting the Cartesian Bloch vector β = (βx, βy, βz) in spherical coordinates as β = (β, θ, φ) Returning to the Bloch sphere shown in Fig 3.2, we define the solution branches by the intersections of the sphere with the y-axis, which fixes the relationships among the components and determines the corresponding angular parameters This Bloch-sphere parametrization provides a concise description of all solutions in terms of β, θ, and φ, from which the relevant physical observables can be readily extracted.
2( −i 1 ). Using this notation, the resulting solution to equation (4.3) can be expressed as the real part of:
Expressed in this manner, a, b, cand d are arbitrary complex quantities.
Returning to the Schrödinger equation in (3.4) and adopting the representation of the vector β in spherical coordinates, one obtains a complete set of solutions for an electron in a constant three-dimensional magnetic field The method yields wavefunctions whose radial and angular parts arise from the spherical components of β, producing a well-defined spectrum of eigenstates for this three-dimensional magnetic-field problem The resulting solutions can be expressed in closed form, providing explicit dependence on the spherical variables and the magnetic-field parameters.
Where f and g are complex quantities that are subject to the normalization condition |f| 2 +|g| 2 = 1.
As was established in section 3.3, the next step in showing that the
To describe the spin state using the Lagrangian in equation (4.2), one seeks a map that renders the solutions of equation (4.4) equivalent to those of equation (4.5) As shown by Wharton, Linck, and Salazar-Lazaro [WLS11], there exist several maps between these two formulations, each valid under a different set of imposed conditions In general, these mappings can be written as χ+ = (x1 + i x2) A* + (x3 − i x4) B*, where χ+ denotes the transformed spin state and x1, x2, x3, x4 are real components of the original spinor, with A* and B* representing complex conjugate amplitudes that encode the mapping.
Where A and B are new complex quantities that are subject to the normalization condition |A| 2 +|B| 2 = 1 In addition, the map in equations (4.6) requires that a=√
2Bg Using these definitions serves to impose the condition: ad
From the map shown in equation (4.6), the complex quantities A and B can take arbitrary values, since any pair that satisfies the definitions of a, b, c, and d is allowed This makes the map in (4.6) many-to-one, so any coupled oscillator solution that satisfies the condition ad maps directly onto a single spin state solution Interestingly, the condition ad also means that the coupled oscillator Lagrangian vanishes, i.e., L^2 = 0.
Before moving forward, we must address a clear issue: the coupled-oscillator solution relies on eight parameters—the real and imaginary parts of a, b, c, and d—while the spin-state solution uses only four parameters, the real and imaginary parts of f and g To resolve this discrepancy, we examine how the imposed constraints reduce the number of free parameters needed to describe each solution and determine the resulting degrees of freedom for the coupled-oscillator and spin-state descriptions.
Table 4.1: Effect of Constraints on Number of Required Free Parameters
Note that since the condition ad (which results in L 2 = 0), is a complex expression it actually imposes two constraints.
From this table it is evident that equation (4.4) requires double the number of free parameters of equation (4.5); to balance the two, three additional free parameters must be added to the spin-state solution Moreover, the parameters of the coupled-oscillator model (a, b, c, and d) can be expressed in terms of both the spin-state parameters (f and g) and the map parameters (A and others).
Within the coupled-oscillator framework, six free parameters are divided between the spin-state solution and the mapping This division treats A and B as hidden variables needed to fully describe the spin state Consequently, the map described by equations (4.6) enables the Lagrangian in equation (4.2) to accurately capture the dynamics of an unmeasured electron spin state in a constant three-dimensional magnetic field.
Time Varying 3D Magnetic Field - Part I (Vectors)
Building on the previous results, we show that the Lagrangian in equation (4.2) can describe the dynamics of an unmeasured electron spin state under a time-varying three-dimensional magnetic field Treating the spin as an internal degree of freedom, the Lagrangian formalism captures how the spin precesses and evolves in response to spatially and temporally varying magnetic fields, providing a coherent quantum-dynamic model for spin evolution This framework links the Lagrangian description to observable spin dynamics, enabling analysis of spin coherence, control, and response in complex 3D magnetic environments.
To do this we will again use Lagrange’s equation to arrive at a set of four coupled equations of motion When expressed in vector notation these equation are:
Where B is a matrix of time dependent β components with exactly the same form as the matrix of constant β components listed in equation (4.1).
Following the current approach, the next step is to establish a mapping between equation (4.7) and the Schrödinger equation (3.4) Although a vector-based analysis could achieve this, employing quaternions offers a more elegant formulation Since quaternions were once widely used in physics but are now less common, this pause also provides a concise review of quaternion algebra and its key properties, helping to illuminate how these characteristics facilitate the correspondence between the two equations.
An Introduction to Quaternions
Quaternions were introduced in 1842 by Irish mathematician Sir William Rowan Hamilton Hamilton aimed to extend complex numbers to describe three-dimensional space, which led him to develop a four-part quaternion with one real component and three imaginary components While a quaternion can be written in several forms, the expression that most closely resembles a complex number is q = q0 + i q1 + j q2 + k q3.
Expressed in this manner, q 0 is the real part of the quaternion whileq 1 , q 2 and q 3 are the imaginary parts of the quaternion.
Complex numbers are built around the relationship, i=√
According to the story, Hamilton’s key insight came during a walk with his wife along Dublin’s Royal Canal in October 1842, near Broome Bridge It was then that he realized the extension he sought could be encapsulated by the quaternion rule i^2 = j^2 = k^2 = ijk = -1, a compact relation that unifies the imaginary units into a single system This fundamental quaternion identity became the cornerstone of quaternions, reshaping how rotations in three dimensions are modeled in mathematics and physics.
i, j, and k are unit quaternions, and the elegance of this relationship impressed Hamilton so much that he immediately carved it into the side of a bridge with a knife, fearing he might die before he could share his discovery with the world Building on this fundamental rule, Hamilton and others went on to describe the algebra of quaternions.
After Hamilton’s discovery, quaternions attracted attention from mathematicians and physicists, and Maxwell even used them in his foundational work on electromagnetic theory However, quaternions proved cumbersome and non-intuitive, and they eventually fell from favor as vector methods replaced them Today, quaternions describe only a small subset of phenomena.
Before discussing the properties of quaternions, it is important to note that the notation used to describe quaternions often resembles the notation used to describe vectors This connection has a historical reason To facilitate the transition from quaternions to vectors, quaternionic notation was incorporated into vector notation One example of this is shown in the notation used to express the quaternion listed in equation (4.8) If q 0 = 0, this quaternion would appear to be a Cartesian vector This is the case because the notation for the Cartesian unit vectors ˆı, ˆ and ˆk were originally borrowed from quaternionic notation.
With this in mind, let us now discuss some of the properties of quaternions. Suppose that a second quaternion is given by, p=p 0 +ip 1 +jp 2 +kp 3 The following is a list of properties of quaternions complied from Hanson [Hans06, Ch 4,
This section cites Kuipers (Kuip99, Ch 5 and Ch 7) and Chapter 7, but the list of quaternion properties is not exhaustive Only those quaternion properties that have proven useful for this work are included.
(1) Quaternion addition is commutative As a result, p+q=q+p.
(2) The sum of two quaternions is a quaternion such that, p+q = (p 0 +q 0 ) +i(p 1 +q 1 ) +j(p 2 +q 2 ) +k(p 3 +q 3 ) (4.10)
(3) Quaternion multiplication is not commutative So in general pq 6=qp.
(4) The product of unit quaternions is defined as: ij =k =−ji jk=i=−kj ki=j =−ik (4.11)
(5) There are two ways to decompose quaternion multiplication using matrices: (a) Left Multiplication: pq
(6) The conjugate of a quaternion is: q ∗ =q 0 −iq 1 −jq 2 −kq 3 (4.14)
(7) The conjugate of a quaternion product is:
(8) The norm of a quaternion is a scalar The square of the quaternion norm is:
(9) The inverse of a quaternion is: q −1 = q ∗
Time Varying 3D Magnetic Field - Part II (Quaternions)
We return to the problem of constructing a map between the coupled oscillator equations of motion in (4.7) and the Schrödinger equation in (3.4) Using quaternion algebra, we begin by defining two quaternions: q = x1 + i x2 + j x3 + k x4, and b = 0 + iβ z − jβ y + kβ x (4.18).
Expressing b in the matrix notation of equation (4.13) is equivalent to the time-dependent matrix of β components in equation (4.1) This equivalence shows that b takes the place of B for right quaternionic multiplications.
By rearranging and changing notation it is possible to express the coupled oscillator equations of motion in (4.7) as, ¨ x+ 2Bx˙ +
By re-expressing this result in quaternions, with q taking the place of x and b taking the place of B under right multiplication, we arrive at the quaternionic equation of motion for the coupled oscillator system: q'' + 2 q' b + q b^2 + b' + ω0^2 = 0.
To arrive at a quaternionic representation of the Schrödinger Equation, we follow a similar procedure to that used for the coupled oscillator equation As was done there, we begin by rearranging the Schrödinger Equation in (3.4) Doing this yields a form suitable for quaternionic treatment and prepares the ground for introducing quaternionic wavefunctions and operators.
Now, suppose that sis the quaternionic representation of the spin state vector |χi. Since |χi= ( χ χ + − ), the spin state quaternion can be expressed as, s=Re(χ + ) +iIm(χ + ) +jRe(χ−) +kIm(χ−) (4.22)
Using χi in place of the original quantity takes us most of the way toward a quaternionic representation of equation (4.21) The remaining step is to express the second term of (4.21) in quaternionic form To accomplish this, we will exploit the mapping i → βã → σ, which provides the bridge to convert that term into a quaternionic expression.
|χi is a complex vector As a result, it can generally be expressed as, i−→ β ã −→σ
Where f 0 ,f 1 , f 2 and f 3 are purely real quantities.
Now suppose that the vector quantityi−→ β ã −→σ
χ can be represented by the quaternionic quantity s_b By a property of quaternionic multiplication, s_b is a quaternion Consequently, it can be written in the general form s_b = g_0 + i g_1 + j g_2 + k g_3 If s_b is the quaternionic representation of i−→ β ã −→σ, its scalar and vector parts encode the corresponding components of that representation.
To be consistent with the definition of s in equation (4.22), it must hold that f0 = g0, f1 = g1, f2 = g2, and f3 = g3 From these equalities, a straightforward multiplication argument shows that the respective parts of i coincide, ensuring full componentwise agreement.
|χi can be represented by the corresponding parts of sb.
Using the correspondence between |χi and salong with the correspondence between i
|χi and sb, it is now possible to express a quaternionic representation of equation (4.21) The resulting quaternionic Schr¨odinger Equation is, ˙ s+sb=−iω 0 s (4.23)
Where i is the unit quaternion.
Using this equation, we can determine whether there is a correspondence between the spin state and the coupled-oscillator solution To do this, we begin by taking the time derivative of equation (4.23) After solving equation (4.23) for ˙s and substituting this expression into the time derivative, we arrive at the following equation: s¨ + 2 ˙s b + s b^2 + ˙b + ω0^2.
Notice that this equation has an identical form to the quaternionic equation of motion for the coupled oscillator system in equation (4.20).
To show the equivalence between the coupled oscillator equation (4.20) and the spin state equation (4.24), we construct a map between the quaternions q and the spin state s Using the quaternion–spin map given in equation (4.6), q can be expressed in terms of the spin-state vector components, yielding q = (χ + A−B ∗ χ ∗ − ) + (χ−A+B ∗ χ ∗ + )j (4.25)
Using this expression it can be shown that q=us, whereu is a constant unit quaternion that is given by the expression: u=Re(A) +iIm(A) +jRe(B)−kIm(B) (4.26)
This result provides a direct map from the coupled-oscillator solution to the spin state Solving for s yields s = u^{-1} q By the quaternion inverse defined in equation (4.17), u^{-1} = u^* / |u|^2 Since |u|^2 corresponds to the normalization condition |A|^2 + |B|^2, it follows that |u|^2 = 1, and therefore s = u^* q.
As a result, the map from the coupled oscillator solution onto the spin state can be expressed as, s=u ∗ q (4.27)
Using the map defined with the associated definition of u, the coupled oscillator equation (4.20) is equivalent to the spin-state equation (4.24), showing that the Lagrangian (4.2) can describe the dynamics of an unmeasured electron spin state in a time-varying three-dimensional magnetic field Returning to the Lagrangian (4.2) and using the notation introduced in this section, we can express it in quaternion form; we begin by expressing the conjugate momentum vector p from (4.1) as a quaternion, and following the same method used for (4.20), the conjugate momentum quaternion takes the form p = ˙q + q b (4.28).
Using this expression for p while replacing xwith q yields the following quaternionic Lagrangian,
Let the conjugate momentum quaternion be expressed in terms of q and u by starting from q = u s to express p in terms of u and s Then, using equation (4.23) and the map in equation (4.27), it follows that the conjugate momentum quaternion is p = −u i ω0 u* q This form enables re-expression of the quaternionic Lagrangian, and by applying the definitions of the quaternion norm and the conjugate of a quaternion product one finds |p|^2 = ω0 |u|^4 |q|^2 Substituting this into the Lagrangian and noting that the quaternion norm is a scalar yields L3 = (1/2) m ω0 (|u|^4 − 1) |q|^2, and since u is a unit quaternion with |u| = 1, it follows that L3 = 0.
Recall that L1 = 0 for both the constant and time-dependent one-dimensional magnetic field configurations, and L2 = 0 for the constant three-dimensional magnetic field arrangement The L3 = 0 result shows that the Lagrangian for the time-dependent three-dimensional magnetic field arrangement is consistent with the other Lagrangians Since none of these conditions were imposed on the system and instead arose from the mapping needed to show correspondence with the spin state, the L = 0 result may point to a deeper underlying truth about these systems.
CHAPTER 5FIRST ORDER LAGRANGIAN FOR SPIN
The Lagrangian for Spin
From the analysis so far, the Lagrangian in equation (4.2), together with its quaternion extension in (4.29), can describe the evolution of an unmeasured electron spin state in a time-varying three-dimensional magnetic field However, the derived equations of motion, listed in (4.3), (4.7), and (4.20), are second-order differential equations, whereas the Schrödinger equation (3.4) is first-order This contrast raises the question of whether a second-order formulation is necessary to capture spin dynamics To address this, one can develop a Lagrangian for electron spin that produces first-order equations of motion, aligning the description with the structure of quantum mechanics while preserving the ability to model the spin's behavior in a dynamic magnetic field.
Recall from the original discussion of the Schr¨odinger equation in (3.4), that the Hamiltonian for spin can be expressed as,H =~(ω 0 I+βãσ) A Lagrangian that incorporates this Hamiltonian is:
Note that when this Lagrangian is stated in terms of the spin Hamiltonian, H, it can also be expressed as,L=hχ|H|χi −~Imhχ|χi.˙
To test whether this Lagrangian correctly describes electron spin, we begin by applying Lagrange’s equation to derive the equations of motion This requires expressing the spin state in terms of a real set of dynamical variables so that the Lagrangian formalism can be used directly By rewriting the spin degrees of freedom as real components, we obtain a system of real equations whose structure reveals whether the spin dynamics are consistent with established spin behavior, such as precession in external fields These steps provide the essential criteria to validate or constrain the proposed Lagrangian for electron spin.
Recall from section 3.3 that the spin state can be expressed as |χi a(t) b(t)
, where a=a R +ia I and b =b R +ib I Using this notation it is then possible to arrive at the following set of coupled equations of motion: ˙ a R = (ω 0 +β z )a I −β y b R +β x b I b˙ R =β y a R +β x a I + (ω 0 −β z )b I ˙ a I =−(ω 0 +β z )a R −β x b R −β y b I b˙ I =−β x a R +β y a I −(ω 0 −β z )b R (5.2)
All four first-order differential equations can be treated as real, and their equivalence to the Schrödinger equation (3.4) is established by decomposing the Schrödinger equation into four real components Repeating the same real-variable decomposition used to express the spin state |χi, one finds that the Schrödinger equation can be recast in a form identical to equations (5.2) Therefore, the Lagrangian shown in equation (5.1) provides a complete description of the interaction between an electron’s spin state and an external magnetic field.
Comparison of First and Second Order Lagrangians
Building on the result from the last section, let us now compare the coupled oscillator Lagrangian in equation (4.2) with the new Lagrangian in equation (5.1).
To aid in this comparison, recall that the two Lagrangians can be expressed as,
First Order Lagrangian (eq 5.1): L=~[hχ|ω 0 I+βãσ|χi −Imhχ|χi]˙ Second Order Lagrangian (eq 4.2): L= 1 2 m(pãp−ω 0 2 xãx)
Where the terms first order and second order refer to the order of the equation’s associated Lagrange’s equations.
By comparing the form of these two equations and their associated Lagrange’s equations (with regard to their solutions), a number of differences present themselves.
Although both formulations yield real equations, the first-order Lagrangian is written with complex quantities, while the second-order Lagrangian is expressed entirely in real quantities It is possible to recast the first-order Lagrangian using only real variables, but this real reformulation is intricate and does not admit an obvious simplification.
In classical mechanics, the Lagrangian is often written as L = T − V, where T is the system’s kinetic energy and V is its potential energy However, this is not the only way to define the classical Lagrangian; in general, a classical Lagrangian is any function that yields the correct equations of motion via the Euler–Lagrange equations Consequently, both first-order and second-order Lagrangians are regarded as classical Lagrangians, though the second-order formulation has a clear classical interpretation while the first-order form does not have an obvious classical analog.
Mapping the solutions of the second-order Lagrangian's equations of motion onto the spin state requires introducing additional hidden parameters In contrast, the first-order Lagrangian yields spin-state solutions directly and does not require extra parameters.
According to Goldstein (Ch 10), the Lagrangian for a one-dimensional harmonic oscillator in the second-order form has a familiar quadratic structure, making the oscillator a natural framework for understanding the system’s underlying dynamics By contrast, the complex form of the first-order Lagrangian lacks an obvious classical analogue, so there isn’t a straightforward classical framework for interpreting its associated dynamics.
(5) According to Goldstein [Gold02, Ch 13], the Klein-Gordon Lagrangian density (in the limit where c= 1), can be expressed as:
By disregarding the middle term, which arises from the spatial component of the Lagrangian density, the second-order Lagrangian adopts a form that closely mirrors this equation This observation points to a natural connection between the second-order Lagrangian framework and the equation under consideration, highlighting how the spatial contribution shapes the overall dynamics.
Relativistic quantum theory currently does not include an equation that mirrors the first-order Lagrangian, leaving unclear how a first-order Lagrangian might be accommodated within its framework.
Building on this work, Wharton [Whar13] shows that a second-order Lagrangian enables extending the classical analogy for the electron spin state to capture more quantum characteristics while expanding the system to include states with arbitrary spin values, a feat not achievable with the first-order Lagrangian.
Taking these differences into account, both Lagrangians offer distinct advantages Notably, the first-order Lagrangian in equation (5.1) can derive the Schrödinger equation directly without invoking a mapping, marking it as a significant result In contrast, the second-order Lagrangian in equation (4.2) is easier to interpret within the classical physics framework Since this work aims to describe the electron spin state using classical physics, the second-order Lagrangian provides a better fit for the analysis.
The Lagrangian is a foundational tool extensively used in both classical and quantum mechanics, providing a unifying framework for describing dynamical systems Building on this framework, we use the classical Lagrangian that describes Foucault’s pendulum to model the dynamics of the electron spin state in an arbitrary magnetic field, revealing a deep connection between pendulum motion and spin evolution This approach bridges classical intuition and quantum spin dynamics, allowing a coherent description of spin behavior in diverse magnetic environments Before discussing the significance of our results, let us first review what we have managed to show so far.
Chapter 2 derives the classical Lagrangian (3.1) describing the dynamics of Foucault’s pendulum, while Chapter 3 uses this Lagrangian to formulate and solve a set of coupled equations of motion By applying the Schrödinger equation (3.4), we obtain a spin‑state solution for an electron in a constant one‑dimensional magnetic field A direct comparison reveals only limited correspondence between the pendulum solutions and the spin state, highlighting the distinct behaviors of classical pendulum dynamics and quantum spin dynamics.
To achieve a clearer correspondence, we revisited the original derivation of the pendulum Lagrangian By modeling the pendulum as two coupled oscillators, we derive a Lagrangian presented in equation (3.11), which encapsulates the essential dynamics of the system within the Lagrangian formalism.
Using the Lagrangian formulation, we showed that the system yields pendulum solutions of the exact required form To push the limits of this correspondence, we extended the analysis to a one-dimensional magnetic field with explicit time dependence With the mapping in equation (3.16), the pendulum’s equations of motion become equivalent to the Schrödinger equation.
Building on the success found using the Lagrangian for two coupled oscillators, we extend the model to four oscillators with three coupling parameters, as encoded by the Lagrangian in equation (4.2) From this Lagrangian we derive and solve a set of four coupled equations of motion Through the map described in equations (4.6), we show that the four-oscillator Lagrangian can describe the dynamics of an unmeasured electron spin state in a constant three dimensional magnetic field.
The final step of the four-oscillator Lagrangian was to reformulate the system in quaternion form In this quaternionic representation, the equations of motion for the four-oscillator system (Eq 4.20) are shown to be equivalent to the quaternionic Schrödinger equation (Eq 4.23) under the mapping described in Eq 4.27 This result implies that the Lagrangian of Eq 4.2 and its quaternionic counterpart in Eq 4.29 can describe the dynamics of an unmeasured electron spin state in a time‑varying three-dimensional magnetic field.
Foucault’s pendulum offers a dual utility: it provides a concrete model for the dynamics of the electron spin state and, more importantly, serves as a conceptual framework for understanding certain quantum aspects of spin In the introduction we identified a set of properties that have historically been used to argue that the electron spin state is inherently non-classical, and this section restates that list to emphasize how these features resist classical description Through this parallel, the pendulum dynamics illuminate how spin precession, coherence, measurement disturbance, and contextuality manifest in quantum spin systems, reinforcing the view that spin behaves in ways that have no classical analogue.
(1) The spin state must be described using complex numbers.
(2) When rotated, the spin state must undergo a 4π (as opposed to the classical 2π) rotation to return to its original state.
(3) The electron’s gyromagnetic ratio is double the classically predicted value.
(4) Measurements of the spin state have discrete outcomes that are predicted by associated probabilities.