# Classical, Quantum and Event-by-Event Simulation of a Stern–Gerlach Experiment with Neutrons

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## Abstract

**:**

## 1. Introduction

## 2. Neutron Experiment

## 3. Newtonian Mechanics

#### 3.1. Model for the Magnetic Field

#### 3.2. Analytically Solvable Cases

#### 3.3. Model Parameters

#### 3.4. Numerical Solution of Equation (3)

#### 3.5. Newtonian Dynamics: Simulation Results for Neutrons

## 4. Quantum-Theoretical Model

#### 4.1. Quantum Theory: Simulation Results

#### 4.2. Quantum Theory: Simplified Model

## 5. Event-by-Event Simulation

- 4.
- If $y\in [{y}_{0},{y}_{1}]$ set $\mathbf{F}=\gamma \phantom{\rule{0.277778em}{0ex}}{B}_{1}{S}^{z}{\mathbf{e}}_{z}-\gamma \phantom{\rule{0.277778em}{0ex}}{B}_{1}{S}^{x}{\mathbf{e}}_{x}$,

- 4.
- If $y\in [{y}_{0},{y}_{1}]$: the
**first**time that the event $\parallel \mathbf{B}\left(\mathbf{x}\right)\parallel >0$ occurs, that is when the particle enters the region where $\parallel \mathbf{B}\left(\mathbf{x}\right)\parallel >0$, use Equation (16) with $\mathbf{s}=\mathbf{S}$ to align the vector $\mathbf{S}$ along the magnetic field ${s}_{\mathbf{b}}\mathbf{B}\left(\mathbf{x}\right)$ and compute $\mathbf{F}=\gamma \phantom{\rule{0.277778em}{0ex}}{B}_{1}{S}^{z}{\mathbf{e}}_{z}-\gamma \phantom{\rule{0.277778em}{0ex}}{B}_{1}{S}^{x}{\mathbf{e}}_{x}$.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Large Static Field B_{0}

## Appendix B. Numerical Solution of Equation (3)

- $\mathbf{v}\leftarrow \mathbf{v}+\tau \mathbf{F}/2m$,
- $\mathbf{x}\leftarrow \mathbf{x}+\tau \mathbf{v}$,
- $\mathbf{S}\leftarrow \mathbf{R}\left(\tau \right)\mathbf{S}$
- $\mathbf{F}=\gamma \phantom{\rule{0.277778em}{0ex}}{B}_{1}{S}^{z}{\mathbf{e}}_{z}-\gamma \phantom{\rule{0.277778em}{0ex}}{B}_{1}{S}^{x}{\mathbf{e}}_{x}$,
- $\mathbf{v}\leftarrow \mathbf{v}+\tau \mathbf{F}/2m$,

**Figure A1.**Histograms of the transverse velocity distribution obtained by solving the classical equations of motion Equation (3) for the model parameters pertaining to imaginary silver particles and for different values of the uniform magnetic field ${B}_{0}$. The initial magnetic moments distributed randomly (see text) (

**a**) ${B}_{0}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**b**) ${B}_{0}=0.1\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**c**) ${B}_{0}=0.01\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**d**) ${B}_{0}=0.001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**e**) ${B}_{0}=0.0001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**f**) ${B}_{0}=0.00001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$.

## Appendix C. Newtonian Dynamics: Imaginary Silver Particles

## Appendix D. Quantum-Theoretical Model

#### Appendix D.1. Momentum Representation

#### Appendix D.2. Textbook Model

#### Appendix D.3. Dimensionless Form

#### Appendix D.4. Initial State

#### Appendix D.5. Simulation Method

#### Appendix D.6. TDPE Solver: Technical Aspects

#### Appendix D.7. Quantum Dynamics: Imaginary Silver Particles

**Figure A2.**Probability distributions $|\langle {v}_{x},{v}_{z}|\mathsf{\Phi}\left({t}^{*}\right)\rangle {|}^{2}$ of the transverse velocity distribution obtained by solving the TDPE Equation (A24) with the initial state given by Equation (A27) and for the model parameters pertaining to imaginary silver particles (${v}_{0}={v}^{*}$). Initially, the variance (dimensionless) $\sigma =0.1$ and the spin state is $\left(\right|\uparrow \rangle +|\downarrow \rangle )/\sqrt{2}$. (

**a**) ${B}_{0}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**b**) ${B}_{0}=0.1\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**c**) ${B}_{0}=0.01\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**d**) ${B}_{0}=0.001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**e**) ${B}_{0}=0.0001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**f**) ${B}_{0}=0.00001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$.

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**Figure 1.**Diagram (not to scale) of a Stern–Gerlach experiment with cold neutrons, performed by Hamelin et al. [10]. After passing through the collimators, most neutrons travel along the y-direction. The cone indicates the directions in which the neutron may, but not necessarily have to, leave the magnetic field region.

**Figure 3.**Histograms of the transverse velocity distribution obtained by the solving the classical equations of motion Equation (3) with the initial magnetic moments distributed randomly (see text) and for different values of the uniform magnetic field ${B}_{0}$. (

**a**) ${B}_{0}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**b**) ${B}_{0}=0.1\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**c**) ${B}_{0}=0.01\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**d**) ${B}_{0}=0.001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$, hard to see but looks similar to a projection of an elongated pacifier; (

**e**) ${B}_{0}=0.0001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**f**) ${B}_{0}=0.00001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$.

**Figure 4.**(

**a**) Histogram of the transverse velocity distribution obtained by solving the classical equations of motion Equation (3) for ${B}_{0}=0$, with the initial magnetic moments distributed randomly (see text). (

**b**) Distribution of the number of particles obtained by integrating the histogram shown in (

**a**) for ${v}_{z}\in [-{v}^{*},{v}^{*}]/100$, as indicated by the gray dashed line in (

**a**).

**Figure 5.**Histogram of the average of the three spin components in the transverse velocity distribution obtained by solving the classical equations of motion Equation (3) with the initial magnetic moments distributed randomly (see text). (

**a**) Average calculated by integrating the spin data for ${v}_{z}\in [-{v}^{*},{v}^{*}]/100$. (

**b**) Average calculated by integrating the spin data for ${v}_{x}\in [-{v}^{*},{v}^{*}]/100$.

**Figure 6.**(

**a**) Same as Figure 4a except that as the particles depart from the source, the variance of the transverse velocity ${\sigma}_{v}=0.28{v}^{*}$. (

**b**) Distribution of the number of particles obtained by integrating the histogram shown in (

**a**) for ${v}_{z}\in [-{v}^{*},{v}^{*}]/100$. The distribution obtained by integrating the same histogram for ${v}_{x}\in [-{v}^{*},{v}^{*}]/100$ looks identical and is therefore not shown.

**Figure 7.**Same as Figure 4 except that the parameters for neutrons have been replaced by the parameters for imaginary silver particles.

**Figure 8.**Probability distribution $|\langle {v}_{x},{v}_{z}|\mathsf{\Phi}({t}^{*}/10)\rangle {|}^{2}$ (${v}_{0}={v}^{*}/10$) obtained by solving the TDPE Equation (A24) with the initial state given by Equation (A27). Initially, the (dimensionless) variance $\sigma =0.1$ and the spin state is $\left(\right|\uparrow \rangle +|\downarrow \rangle )/\sqrt{2}$. (

**a**) ${B}_{0}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**b**) ${B}_{0}=0.1\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**c**) ${B}_{0}=0.01\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**d**) ${B}_{0}=0.001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**e**) ${B}_{0}=0.0001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**f**) ${B}_{0}=0.00001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$.

**Figure 9.**(

**a**) Probability distribution $|\langle {v}_{x},{v}_{z}|\mathsf{\Phi}({t}^{*}/10)\rangle {|}^{2}$ (${v}_{0}={v}^{*}/10$) of the transverse velocity distribution obtained by solving the TDPE Equation (A24) with the initial state given by Equation (A27) and ${B}_{0}=0$. Initially, the (dimensionless) variance $\sigma =0.1$ and the spin state is $|\uparrow \rangle $. (

**b**) Same as (a) except that the initial spin state is $|\downarrow \rangle $.

**Figure 10.**(

**a**) One-dimensional probability distributions $p\left({v}_{x}\right)={\left|{\mathsf{\Phi}}_{+1}({v}_{x},{v}_{z}=0,{t}^{*}/10)\right|}^{2}$ (u, solid line) and $p\left({v}_{x}\right)={\left|{\mathsf{\Phi}}_{-1}({v}_{x},{v}_{z}=0,{t}^{*}/10)\right|}^{2}$ (d, dotted line), extracted from the data shown in Figure 9a. Except for $|{v}_{x}/{v}_{0}|\le 0.2$, the difference between two distributions is too small to be visible in the plot. (

**b**) One-dimensional probability distributions $p\left({v}_{z}\right)={\left|{\mathsf{\Phi}}_{+1}({v}_{x}=0,{v}_{z},{t}^{*}/10)\right|}^{2}$ (u, solid line) and $p\left({v}_{z}\right)={\left|{\mathsf{\Phi}}_{-1}({v}_{x}=0,{v}_{z},{t}^{*}/10)\right|}^{2}$ (d, dotted line), extracted from the data shown in Figure 9a. The probability distributions $|{\mathsf{\Phi}}_{-1}({v}_{x}=0,{v}_{z},{t}^{*}/10){|}^{2}$ is too small to be visible in the plot. Except for $|{v}_{x}/{v}_{0}|\le 0.2$, the difference between two distributions is too small to be visible in the plot. For presentation purposes, each distribution is normalized such that its maximum is one. As in Figure 9, ${v}_{0}={v}^{*}/10$.

**Figure 11.**Histograms of the transverse velocity distribution obtained by event-by-event simulation. The classical equations of motion (Equation (3)) are modified to include a one-time projection of the spin vector $\mathbf{S}$ along the direction of the magnetic field using the procedure described in the text. The initial magnetic moments of the particles are distributed randomly. The variance of the transverse velocity ${\sigma}_{v}=0.014{v}^{*}$. (

**a**) ${B}_{0}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**b**) ${B}_{0}=0.1\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**c**) ${B}_{0}=0.01\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**d**) ${B}_{0}=0.001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**e**) ${B}_{0}=0.0001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$; (

**f**) ${B}_{0}=0.00001\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$.

**Table 1.**Overview of the shapes of the transverse velocity distributions obtained by computer simulation of three different descriptions of the SG experiment with cold neutrons.

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**MDPI and ACS Style**

De Raedt, H.; Jin, F.; Michielsen, K.
Classical, Quantum and Event-by-Event Simulation of a Stern–Gerlach Experiment with Neutrons. *Entropy* **2022**, *24*, 1143.
https://doi.org/10.3390/e24081143

**AMA Style**

De Raedt H, Jin F, Michielsen K.
Classical, Quantum and Event-by-Event Simulation of a Stern–Gerlach Experiment with Neutrons. *Entropy*. 2022; 24(8):1143.
https://doi.org/10.3390/e24081143

**Chicago/Turabian Style**

De Raedt, Hans, Fengping Jin, and Kristel Michielsen.
2022. "Classical, Quantum and Event-by-Event Simulation of a Stern–Gerlach Experiment with Neutrons" *Entropy* 24, no. 8: 1143.
https://doi.org/10.3390/e24081143