# Asymmetric pulse effects on pair production in chirped electric fields

Neng-Zhi Chen  Key Laboratory of Beam Technology of the Ministry of Education,
and College of Nuclear Science and Technology, Beijing Normal University,
Beijing 100875, China  Orkash Amat  Key Laboratory of Beam Technology of the
Ministry of Education, and College of Nuclear Science and Technology, Beijing
Normal University, Beijing 100875, China  Li-Na Hu  Key Laboratory of Beam
Technology of the Ministry of Education, and College of Nuclear Science and
Technology, Beijing Normal University, Beijing 100875, China  Hong-Hao Fan
Key Laboratory of Beam Technology of the Ministry of Education, and College of
Nuclear Science and Technology, Beijing Normal University, Beijing 100875,
China  Bai-Song Xie<EMAIL_ADDRESS>Key Laboratory of Beam Technology of
the Ministry of Education, and College of Nuclear Science and Technology,
Beijing Normal University, Beijing 100875, China  Institute of Radiation
Technology, Beijing Academy of Science and Technology, Beijing 100875, China

###### Abstract

We investigate the effects of the asymmetric pulse shapes on electron-positron
pair production in three distinct fields: chirp-free, small frequency chirp,
and large frequency chirp fields via the real-time Dirac-Heisenberg-Wigner
formalism. Our findings reveal the disappearance of interference effects with
shorter falling pulse length, and the peak is concentrated on the left side of
the momentum spectrum. As the falling pulse length extends, an incomplete
multi-ring structure appears in the momentum spectrum. The number density of
particles are very sensitive to the asymmetry of the pulse. With a long
falling pulse, the number density can be significantly enhanced by over four
orders of magnitude when certain frequency chirps are utilized. These results
highlight the impact of the effective dynamically assisted mechanism and the
frequency chirp on pair creation.

###### pacs:

12.20.Ds, 11.15.Tk

##  I Introduction

Electron-positron ${(e^{-}e^{+})}$ pair production in the vacuum under strong
electromagnetic fields stands as one of the most intriguing phenomena in
relativistic quantum physics dirac1928quantum ; sauter1931verhalten ;
anderson1933positive ; heisenberg1936folgerungen ; schwinger1951gauge ;
xie2017electron . This effect has yet to be directly observed in the
laboratory. The reason is that the critical field strength
$E_{\mathrm{cr}}=m^{2}c^{3}/e\hbar\approx 1.3\times
10^{16}~{}\rm{\mathrm{V}/\mathrm{cm}}$ is too high, which corresponds to the
laser intensity of about $4.3\times 10^{29}~{}\mathrm{W/cm^{2}}$ (where $m$
and $-e$ is the electron mass and charge, respectively). Such an intensity
level is presently unattainable. Nevertheless, X-ray Free Electron Laser
systems approach near-critical field strengths of ${E/{E}_{\mathrm{cr}}\approx
0.01-0.1}$ ringwald2001pair . Recent advancements in high-intensity laser
technology heinzl2009exploring ; pike2014photon  provide prospects for
potential experimental validation in the near future.

The influence of different pulse profiles on pair production has been a
subject of extensive research. Hebenstreit et al. noted a significant
sensitivity of momentum spectra to external field parameters using short-pulse
lasers PhysRevLett.102.150404 . Furthermore, in spatially inhomogeneous pulse
fields, the pair production process is accompanied by particle self-focusing
effects PhysRevLett.107.180403 . Schützhold et al. introduced a dynamically
assisted Schwinger mechanism, which combines low-frequency strong fields with
high-frequency weak fields schutzhold2008dynamically , substantially
amplifying the particle production rate. Recently, the impact of frequency
chirp effects on particle momentum spectra and number density in time-
dependent electric field dumlu2010schwinger ; olugh2019pair ;
bai2022enhancement ; wang2023enhancement  has attracted increased attention.
Specifically, research has focused on enhancing particle creation in both
time-dependent monochromatic and bichromatic laser fields subjected to
frequency chirp abdukerim2017enhanced ; ababekri2020chirp ; wang2021effect ;
gong2020electron ; mohamedsedik2021schwinger ; LiLieJuan2021 , and exploring
asymmetric pulse shapes in single-color fields oluk2014electron ;
olugh2020asymmetric . Numerous studies aim to enhance pair production through
various field combinations.

In this study, we explore vacuum pair production, considering a range of chirp
parameters and asymmetric pulse shapes in the electric field. Our approach is
based on the Dirac-Heisenberg-Wigner (DHW) formalism, which can handle diverse
chirp and asymmetric envelope fields. We observe that the momentum spectrum
exhibits a high sensitivity to chirp, revealing interference effects for
varying pulse shapes and chirp parameters. The particle number density is in
general enhanced with increasing chirp. Specifically, in instances of extended
falling pulses, the effect of chirp results in an enhancement of the particle
number density by up to four orders of magnitude. Throughout our study, we
adopt natural units ($\hbar=c=1$) and express all quantities in terms of the
electron mass $m$.

The paper is structured as follows: In Sec. II, we provide a brief overview of
the DHW formalism applied in this study. In Sec. III, we introduce the model
for background fields. Within Sec. IV, we present numerical outcomes
concerning the momentum spectrum and analyze the underlying physics. In Sec.
V, we focus on numerical findings related to the number density. Finally, in
Sec. VI, we provide a summary and discussion.

##  II DHW formalism

The DHW formalism is an approach used to describe quantum phenomena within a
system, utilizing the Wigner function to represent the relativistic phase
space distribution. We employ the widely adopted DHW formalism for
investigating vacuum pair production within strong background fields
bialynicki1991phase ; hebenstreit2010schwinger ; hebenstreit2011particle ;
kohlfurst2020effect ; ababekri2020chirp ; li2021study ; mohamedsedik2023phase
; PhysRevD.108.056011 ; PhysRevD.107.116010 . Since the specific derivation of
DHW has been explained in prior studies kohlfurst2015electron ;
vasak1987quantum , this work focuses on presenting the fundamental concepts
and essential aspects of this method.

For convenience, we start with the system’s gauge-invariant density operator,

$\hat{\mathcal{C}}_{\alpha\beta}\left(r,s\right)=\mathcal{U}\left(A,r,s\right)\
\left[\bar{\psi}_{\beta}\left(r-s/2\right),\psi_{\alpha}\left(r+s/2\right)\right],$
(1)

here, the electron’s spinor-valued Dirac field is denoted by
${\psi}_{\beta}\left({x}\right)$, where $r$ represents the center of mass and
$s$ denotes the relative coordinates. The introduction of the Wilson line
factor, $\mathcal{U}(A,r,s)$, aims to maintain the gauge invariance of the
density operator. This factor’s properties are determined by the elementary
charge $e$ and the background gauge field $A$.

An essential component of the DHW formalism is the covariant Wigner operator,
which is obtained by Fourier transforming the density operator Eq.(1),

$\hat{\mathcal{W}}_{\alpha\beta}\left(r,p\right)=\frac{1}{2}\int d^{4}s\
\mathrm{e}^{\mathrm{i}ps}\ \hat{\mathcal{C}}_{\alpha\beta}\left(r,s\right).$
(2)

Upon calculating the vacuum expectation value of the Wigner operator, we
obtain the Wigner function,

$\mathbb{W}\left(r,p\right)=\langle\Phi|\hat{\mathcal{W}}\left(r,p\right)|\Phi\rangle.$
(3)

To derive the time-evolution equation, we rely on the equal-time Wigner
function

$\mathbbm{w}(\mathbf{x},\mathbf{p},t)=\int\frac{dp_{0}}{2\pi}\mathbb{W}(r,p).$
(4)

The Wigner function can be decomposed into a complete basis set of Dirac
matrices, resulting in 16 covariant real Wigner components

$\mathbbm{w}=\frac{1}{4}\left(\mathbbm{1}\mathbbm{s}+\textrm{i}\gamma_{5}\mathbbm{p}+\gamma^{\mu}\mathbbm{v}_{\mu}+\gamma^{\mu}\gamma_{5}\mathbbm{a}_{\mu}+\sigma^{\mu\nu}\mathbbm{t}_{\mu\nu}\right),\
$ (5)

where $\mathbbm{s}$, $\mathbbm{p}$, $\mathbbm{v}_{\mu}$, $\mathbbm{a}_{\mu}$
and $\mathbbm{t}_{\mu\nu}$ denote scalar, pseudoscalar, vector, axial vector
and tensor, respectively. According to the Refs. hebenstreit2010schwinger ;
hebenstreit2011particle ; kohlfurst2015electron , the motion equation for the
Wigner function is:

$D_{t}\mathbbm{w}=-\frac{1}{2}\mathbf{D}_{\mathbf{x}}[\gamma^{0}\bm{\gamma},\mathbbm{w}]+\mathrm{i}m[\gamma^{0},\mathbbm{w}]-\mathrm{i}\mathbf{P}\\{\gamma^{0}\bm{\gamma},\mathbbm{w}\\},$
(6)

where $D_{t}$, $\mathbf{D}_{\mathbf{x}}$ and $\mathbf{P}$ are represented as
pseudodifferential operators.

By embedding the motion Eq. (5) into Eq. (6), we obtain a system of partial
differential equations for the 16 Wigner components. Besides, for spatial
homogeneous time-dependent electric fields we can use the characteristic
method blinne2016comparison  to replace the dynamical momentum ${\mathbf{p}}$
with the canonical momentum ${\mathbf{q}}$ via
${\mathbf{q}}-e{\mathbf{A}}(t)$, Thus, the system of partial differential
equations for the 16 Wigner components can be simplified to a set of 10
ordinary differential equations:

${\mathbbm{w}}=({\mathbbm{s}},{\mathbbm{v}}_{i},{\mathbbm{a}}_{i},{\mathbbm{t}}_{i})\,,\quad{\mathbbm{t}}_{i}:={\mathbbm{t}}_{0i}-{\mathbbm{t}}_{i0}\,.$
(7)

Due to the complexity of the motion equation of the Wigner function, we do not
present a detailed derivation here, but it is available in the cited Refs.
hebenstreit2011particle ; kohlfurst2015electron . The relevant non-zero vacuum
initial values are as follows:

${\mathbbm{s}}_{vac}=\frac{-2m}{\sqrt{{\mathbf{p}}^{2}+m^{2}}}\,,\quad{\mathbbm{v}}_{i,vac}=\frac{-2{p_{i}}}{\sqrt{{\mathbf{p}}^{2}+m^{2}}}\,.$
(8)

Hereafter, we denote the scalar Wigner function $f(q,t)$ using the single-
particle momentum distribution function,

$f({\mathbf{q}},t)=\frac{1}{2\Omega(\mathbf{q},t)}(\varepsilon-\varepsilon_{vac}),$
(9)

where
$\Omega=\sqrt{m^{2}+{\mathbf{p}}^{2}}=\sqrt{m^{2}+(\mathbf{q}-e\mathbf{A}(t))^{2}}$
is the total energy of particle,
$\varepsilon=m{\mathbbm{s}}+p_{i}{\mathbbm{v}}_{i}$ represents the phase space
energy density, $m$ represents the electron mass.

To accurately compute the single-particle momentum distribution
$f({\mathbf{q}},t)$, we refer to Ref. blinne2016comparison . We introduce an
auxiliary three-dimensional vector $\mathbf{v}(\mathbf{q},t)$ to facilitate
the calculation:

$\mathbf{v}(\mathbf{q},t):={\mathbbm{v}}_{i}(\mathbf{p}(t),t)-(1-f({\mathbf{q}},t)){\mathbbm{v}}_{i,vac}(\mathbf{p}(t),t)\,.$
(10)

Consequently, by solving the following ordinary differential equations, the
single-particle momentum distribution function $f(q,t)$ can be obtained:

$\begin{array}[]{l}\dot{f}=\frac{e\mathbf{E}\cdot\mathbf{v}}{2\Omega},\\\
\dot{\mathbf{v}}=\frac{2}{\Omega^{3}}[(e\mathbf{E}\cdot\mathbf{p})\mathbf{p}-e\mathbf{E}\Omega^{2}](f-1)-\frac{(e\mathbf{E}\cdot\mathbf{v})\mathbf{p}}{\Omega^{2}}-2\mathbf{p}\times\mathbbm{a}-2m\mathbbm{t},\\\
\dot{\mathbbm{a}}=-2\mathbf{p}\times\mathbf{v},\\\
\dot{\mathbbm{t}}=\frac{2}{m}[m^{2}\mathbf{v}+(\mathbf{p}\cdot\mathbf{v})\mathbf{p}],\end{array}$
(11)

where
$f(\mathbf{q},-\infty)=0\,,\,\mathbf{v}(\mathbf{q},-\infty)=\mathbbm{a}(\mathbf{q},-\infty)=\mathbbm{t}(\mathbf{q},-\infty)=0$
are initial conditions, dot represents a total time derivative, $\mathbf{v}$
denotes current density, $\mathbbm{a}$ is spin density, $\mathbf{E}$ is
electric field, ${\mathbf{p}}$ represents kinetic momentum, $\mathbf{q}$
denotes canonical momentum, $e$ is the charge of particle, i.e., $|e|$ and
$-|e|$ for positron and electron, respectively, $\mathbf{A}(t)$ is the vector
potential of the external field.

Ultimately, the number density of created pairs, defined at the asymptotic
limit as $t\rightarrow+\infty$, can be determined by integrating the
distribution function $f(\mathbf{q},t)$:

$n=\lim_{t\to+\infty}\int\frac{d^{3}q}{(2\pi)^{3}}f(\mathbf{q},t)\,.$ (12)

##  III The external field form

We study the pair production in asymmetric frequency-chirped electric fields.
The external electric field we consider is presented as follows:

$\mathbf{E}(t)=E_{0}\left[\exp\left(-\frac{t^{2}}{2{\tau_{1}}^{2}}\right)H(-t)+\exp\left(-\frac{t^{2}}{2{\tau_{2}}^{2}}\right)H(t)\right]\cos\left(bt^{2}+\omega
t\right),$ (13)

where $E_{0}$ represents the electric field amplitude, $\tau_{1}$ and
$\tau_{2}$ denote the rising and falling pulse duration, H($t$) stands for the
Heaviside step function, $\omega$ corresponds to the oscillation frequency of
the electric field, $b$ is the chirp parameter. We introduce a time-dependent
effective frequency: $\omega_{eff}=\omega+bt$.

Notably, the electric field described by Eq. (13) is solely time-dependent and
can be regarded as a standing wave formed by two laser beams with different
pulse widths and opposite propagation directions. In our study of
${e^{-}e^{+}}$ pair production, we define the following electric field
parameters:

$E_{0}=0.1E_{\mathrm{cr}},\,\omega=0.6\mathrm{~{}m},\,\tau_{1}=10/\mathrm{m},\,\tau_{2}=k\tau_{1},$
(14)

where $E_{\mathrm{cr}}=m^{2}c^{3}/e\hbar\approx 1.3\times
10^{16}\mathrm{V/cm}$ stands as the Schwinger critical field strength, $k$
denotes the ratio of the falling pulse duration to the rising pulse duration,
adjusting the field’s asymmetry. In Fig. 1, we illustrate the influence of
time-dependent electric field over time for different chirp pulse parameters
$b$ and ratio parameter $k$.

Figure 1: Displaying asymmetric time-dependent electric field with ratio
parameter $k=0.5$, $k=1$, $k=2$ and $k=3$. The red dashed line, purple dashed
line and solid blue line are corresponding to the field with chirp parameter
$b=0m^{2}$, $b=0.001m^{2}$ and $b=0.012m^{2}$, respectively. The chosen
parameters are $E_{0}=0.1E_{\mathrm{cr}}$, $\omega=0.6m$, $\tau_{1}=10/m$ and
$\tau_{2}=k\tau_{1}$.

In the study of vacuum pair creation, the Keldysh parameter holds significant
importance and is defined as follows:

$\gamma=m\omega/eE,$ (15)

here, $E$ represents the field strength of the background field. The tunneling
effect and multiphoton absorption can be investigated by considering
$\gamma\ll 1$ and $\gamma\gg 1$, respectively. In our research, the Keldysh
parameter is not significantly larger than 1, indicating that the creation of
particles involves multiphoton absorption and tunneling effects. The pair
production process primarily takes place within the time interval
$-\tau<t<\tau$, we select the chirp parameter to fall within the standard
range, ensuring that $bt$ maximum value is approximately equal to $\omega$.
The linear chirp parameter $b$ can be expressed as $b=\alpha\omega/\tau$,
where $0\leq\alpha\leq 1$ with the maximum chirp value to be $b=0.012m^{2}$.

##  IV Momentum spectra of the produced particles

This section explores the influence of asymmetric pulse shapes on momentum
spectra across varying fields, encompassing chirp-free, small frequency chirp,
and large frequency chirp configurations.

###  IV.1 chirp-free $b=0$

The impact of the asymmetric pulse on particle momentum spectra within the
chirp-free field is illustrated in Fig. 2, exploring a range of $k$ values
from $0$ to $1$. The momentum spectrum is sensitive to the electric field
asymmetry. At $k=1$, the momentum spectrum is symmetric and peaked at the
origin, exhibiting mild oscillations as depicted in Fig. 2(a). These
oscillations arise from interference among separate complex conjugate pairs of
turning points, detailed in Ref. dumlu2010schwinger . When $k=0.7$, the peak
has a slight increase with momentum concentrating around $-0.4m$ as shown in
Fig. 2(b). The breakdown of symmetry is a direct consequence of the asymmetric
nature of the pulse. When setting $k$ to 0.5, the momentum spectra split in
both positive and negative $q_{x}$ directions, resulting in the observation of
two distinct maxima in the momentum spectrum in Fig. 2(c). This pronounced
asymmetry in the momentum spectrum resembles the effects introduced by carrier
phase, as studied in Ref. PhysRevLett.102.150404 . For $k=0.3$, the minor peak
in the negative $q_{x}$ region grows into a major peak; see Fig. 2(d). This
effect is similar to the frequency-chirp effect studied in Ref. olugh2019pair
. Finally, the maximum of the momentum spectrum rises from $4.92\times
10^{-6}(k=1)$ to $6.82\times 10^{-6}(k=0.3)$.

Figure 2: Momentum spectra of particles in the $(q_{x},q_{y})$ plane with
chirp-free $(b=0)$. The falling-pulse duration $\tau_{2}=k\tau_{1}$ is
compressed, with $k$ varying between $0$ and $1$. The chosen parameters for
the electric field are $E_{0}=0.1E_{\mathrm{cr}}$, $\omega=0.6m$, and
$\tau_{1}$ = $10/m$.

For small values of $k$, the role of pulse asymmetry is similar to that of the
carrier phase. In the presence of $E(t)$, the produced pairs are continuously
accelerated, and particle momentum is mainly determined by their creation time
orthaber2011momentum ; kohlfurst2018phase . Particles are created with zero
longitudinal momentum at the earlier time $t_{0}$. After $t_{0}$, they
experience extended acceleration, resulting in heightened longitudinal
momentum. Typically, the majority of pairs emerge during instances
corresponding to local maxima of the field. Subsequently, these produced pairs
undergo acceleration due to the electric field, and the gained momenta are
olugh2019pair .

$q=\int_{t_{0}}^{t}e\mathbf{E}(t^{\prime})\mathrm{d}t^{\prime}=e\mathbf{A}\left(t_{0}\right)-e\mathbf{A}(t).$
(16)

The particle final momentum is determined solely by the vector potential at
the time of its creation, as the vector potential asymptotically approaches
zero as $t\longrightarrow\infty$. For instance, the peak observed in Figure
2(a) when $k=1$ at $q\left(t_{0}\right)=0$ can be attributed to the dominant
peak in the electric field at $t=0$. As time progresses, the electric field
$\mathbf{E}(t)$ decreases, leading to a reduction in the number of particles
produced. However, the vector potential $\mathbf{A}(t)$ increases
simultaneously, resulting in these particles undergoing stronger acceleration.
Consequently, the positions of the peaks and the patterns in the momentum
spectrum are influenced by the pulse shape.

Figure 3: Same as Fig. 2 except that the falling pulse length
$\tau_{2}=k\tau_{1}$ becomes longer with $k\geq 1$.

Moreover, with an increased pulse length ratio $(k\geq 1)$, the momentum
spectra of particles in the $(q_{x},q_{y})$ plane, as shown in Fig. 3, exhibit
a progressive decrease in the central distribution and a gradual decrease in
the peak values. Both of these phenomena are consequences of the intensifying
asymmetry of the background electric field. A significant observation is the
emergence of ring-like structures in the momentum spectrum. This
characteristic pattern is indicative of multiphoton pair production. Since the
predicted particle canonical momentum is

$\left|\mathbf{q}^{*}\right|=\sqrt{\left(\frac{n\omega}{2}\right)^{2}-m_{*}^{2}},$
(17)

where $m_{*}$ and $n$ are the effective mass and absorption photon number
kohlfurst2014effective . The inner ring corresponds to the absorption of four
photons, while the incomplete outer ring structure corresponds to the
absorption of five photons.

###  IV.2 Small Frequency chirp parameter $b=0.002$

Considering a small frequency chirp parameter $(b=0.002)$, Fig. 4 presents
momentum spectra corresponding to shorter pulse length ratios. Specifically,
the peak of the momentum spectrum concentrates in the center with a value of
$5.82\times 10^{-6}$ as depicted in Fig. 4(a). This peak exhibits a slight
increase compared to conditions without frequency chirp effects. Notably, the
distortion of the momentum spectrum and the disruption of symmetry about the
$q_{x}$ axis can be attributed to frequency chirp. As $k$ decreases further,
two distinct momentum peaks emerge, with the left peak continuing to increase,
while the right peak gradually diminishes. When $k$ is set to $0.7$, a minor
increase in the momentum distribution in the negative $q_{x}$ region is
observed, as shown in Fig. 4(b). This phenomenon is associated with pulse
asymmetry. This can be attributed to the shorter falling-pulse duration
$k\tau_{1}$, leading to more particles being accelerated towards the negative
$q_{x}$ direction. At $k=0.5$, the momentum spectrum distinctly splits into
two peaks, extending towards both positive and negative $q_{x}$ directions.
The peak in the negative $q_{x}$ region is smaller, as can be seen in Fig.
4(c). As $k$ decreases to 0.3, the peak in the negative $q_{x}$ region
surpasses that in the positive $q_{x}$ region, as shown in Fig. 4(d).
Furthermore, the oscillation cycles are few for shorter pulses, making the
multiphoton pair production signal less clear in the spectrum due to the
influence of the smaller pulse length $\tau$ on the Keldysh parameter
$\gamma$.

Figure 4: Same as Fig. 2 except for the small frequency chirp parameter with
$b=0.002$.

We consider the falling-pulse extension $(k\geq 1)$, as illustrated in Fig. 5.
For an extended falling-pulse, the peak of the momentum spectrum, situated at
$(0,0)$, increases with $k$. At $k=5$, additional incomplete ring structures
emerge due to chirp effects. This can be attributed to the electric field
sufficient duration and direction change during pair creation as pulse length
$k\tau_{1}$ increases. Hence, particles might be accelerated variably,
resulting in a spectrum ring structure. As pulse duration escalates with $k$,
the oscillation cycles within the Gaussian envelope amplify, leading to more
photons contributing to pair production.

Figure 5: Same as Fig. 3 except for the small frequency chirp parameter with
$b=0.002$.

###  IV.3 Large Frequency chirp parameter $b=0.01$

Figure 6: Same as Fig. 2 except for the large frequency chirp parameter with
$b=0.01$.

Considering a large frequency chirp parameter, $b=0.01$, we delineate the
momentum spectra for short pulses within the range $0<k\leq 1$ in Fig. 6. In
the symmetric case $(k=1)$, the momentum spectrum moves in both positive and
negative $q_{x}$ directions, resulting in a split of the momentum spectrum,
leading to the observation of three maxima, accompanied by an asymmetrical
distribution, as shown in Fig. 6(a). For $k=0.7$, the momentum spectrum
becomes more focused, accompanied by a decrease in the peak value. When $k$ is
set to $0.5$, the momentum spectrum takes on a jellyfish-like shape. A
substantial number of particles are concentrated at the head of the jellyfish-
like momentum spectrum, while the tail exhibits a lower particle distribution.
At $k=0.3$, the jellyfish-like momentum distribution displays a different tail
orientation compared to the case when $k=0.5$. As the pulse length ratio $k$
decreases, the number of oscillations within the envelope becomes exceedingly
limited, thereby restricting the increase in effective frequency
$\omega_{eff}$. Therefore, Fig. 6 (d) is similar to that in Fig. 2 and 4 (d).

Figure 7: Same as Fig. 3 except for the large frequency chirp parameter with
$b=0.01$.

For a large frequency chirp parameter, $b=0.01$, we present the momentum
spectra for an extended pulse length ratio $(1\leq k\leq 5)$ in Fig. 7. As the
falling-pulse duration $k\tau_{1}$ increases, the effect of the chirp
frequency is magnified. At $k=2$, intriguing interference patterns are
observed in Fig. 7 (b). The extended duration of the falling pulse induces
extensive interference, attributable to a significant number of oscillation
periods within the envelope. When $k=3$, partial circular interference
patterns are observed in Fig. 7(c). With increasing pulse length $k\tau_{1}$,
the electric field persists long enough to alter its direction during the pair
creation process. This results in a ring structure of the spectrum. The
asymmetry of the pulse duration contributes to the emergence of these partial
circular patterns. Finally, for $k=5$, these circular structures accumulate
layer by layer, as depicted in Fig. 7(d). Particles are primarily localized at
the center of the circular rings in momentum spectrum. As the ring radius
expands, the momentum distribution diminishes.

##  V Number density of pair production

Figure 8: Number density of particles in asymmetric electric fields with
varying pulse length ratio values $k$, measured in units of
$\lambda_{c}^{-3}$. The chirp parameters $b$ considered in this research are
$0,0.002,0.005,0.007,0.01$, and $0.12$. The electric field parameters chosen
are $E_{0}=0.1E_{\mathrm{cr}}$, $\omega=0.6m$, and $\tau_{1}$= $10/m$.

In this section, we present the number density of particles in chirped
electric fields with the pulse ratio $k$ varying as depicted in Fig. 8. For
the chirp-free field $(b=0)$, we observe a decrease in the particle number
density as the pulse length ratio $k$ increases. Specifically, the number
density drops from $8.31\times 10^{-8}$ for $k=0.3$ to $9.50\times 10^{-9}$
for $k=5$. As $k$ decreases, the field consists of strong pulses for $t<0$ and
weak but with a wider frequency component (in terms of Fourier decomposition)
pulses for $t>0$ . These two semi-pulses, operating at different time scales,
serve as an effective dynamically assisted mechanism, leading to an increase
in the number density. A similar phenomenon is observed in asymmetric linearly
polarized pulses in Figs. 7 and 8 from the Ref. olugh2020asymmetric .

For a small chirp parameter $(b=0.002)$, we observe an initial decline in the
number density of produced particles as the pulse ratio $k$ increases. This
decrease reaches a minimum at $k=1$, corresponding to a value of $1.96\times
10^{-8}$. When $k>1$, the number density begins to rise. The extension of the
pulse duration is accompanied by an increase in the number of oscillation
cycles, thereby enhancing the multiphoton mechanism. Consequently, number
density of particles rapidly increases for larger pulse ratio $k$.
Interestingly, the chirp does not necessarily enhance the number density when
$b$ is small. This nonmonotonic relationship between the number density and
the chirp $b$ is mainly attributed to the temporal structure of the external
field.

For a larger chirp parameter, the number density of particles generated
initially decreases and then increases with the pulse length ratio $k$. With
decreasing values of $k$, the dynamically assisted mechanism becomes
progressively more effective, thereby enhancing the number density.
Conversely, for larger values of $k$, the pulse duration extends, resulting in
an increased number of oscillation cycles within the Gaussian envelope.
Simultaneously, the effective frequency $\omega_{eff}$ increases.
Consequently, more photons contribute to pair production through the
multiphoton absorption mechanism. For instance, in an electric field with
chirp parameter $b=0.012$, the number density reaches its minimum when $k$ is
$0.5$. The two mechanisms of enhanced pair production compete, resulting in
the lowest number density.

In conclusion, reducing the falling pulse length in the chirped field can
potentially increases the number density by up to fivefold. Conversely,
extending the falling pulse can enhance the number density by one to four
orders of magnitude. Therefore, among asymmetric electric fields with varying
chirp, elongating the falling pulse emerges as the most effective strategy for
enhancing particle production.

##  VI summary and discussion

In this study, we investigate the effects of the asymmetric pulse shape on the
momentum spectrum of created ${e^{-}e^{+}}$ pairs in strong electric fields.
We considered three distinct chirp fields: chirp-free, small frequency chirp,
and large frequency chirp fields. Utilizing the DHW formalism, we analyze the
momentum spectrum of the generated particles. Keeping the rising pulse
$\tau_{1}$ constant while modifying the falling pulse $\tau_{2}=k\tau_{1}$,
leading to two asymmetric scenarios: pulse compression $(0<k<1)$ and pulse
extension ($k>1$). As the falling pulse $k\tau_{1}$ compresses, interference
effects gradually diminish with the decreasing pulse length ratio $k$.
Shortening the pulse length induces a shift and split of the peaks.
Conversely, when the falling pulse $\tau_{2}$ extends, an incomplete multiring
structure appears for different chirped fields. The signal of multiphoton pair
production becomes pronounced in the chirp-free field, while interference
effects become noticeable in chirped electric fields.

We also explore the impact of asymmetric falling pulse on the number density.
The particle number density is found to be highly sensitive to both the
asymmetry and chirp parameters of the pulse. When the falling pulse length is
compressed, the number density could increase by up to fivefold. Importantly,
an elongated falling pulse leads to a significant enhancement in the number
density of produced pairs by more than four orders of magnitude. The number
density is influenced by the effective dynamically assisted mechanism and the
frequency chirp. For shorter pulse $(k<1)$, we observe that the traditional
standard multiphoton pair production weakens due to the Keldysh parameter
$\gamma=m\omega/eE$ being modified by another time scale $\tau$ of the pulse
duration olugh2020asymmetric . For very small $\tau$, this implies that the
oscillation number of the field encompasses fewer cycles and subcycles, making
it not strictly a complete multiphoton process. However, in this case, the
number density of pairs can be increased remarkably due to the dynamically
assisted mechanism. On the other hand, for the elongated pulse $k>1$, as $k$
increases, pair production is dominated by the multiphoton mechanism with
chirping.

The influence of chirp frequency on momentum spectrum and number density is
relatively minor during pulse compression. The number density increases
rapidly with chirp parameters in the case of pulse extension. These findings
contribute to understanding the impact of key external parameters, namely
pulse length and chirp parameters, and provide insight into the structure of
the external pulse. While these results reveal useful information about
electron-positron pair production in various chirped fields, this study is
confined to multiphoton pair production. Further research is essential to
investigate the effects of asymmetric pulse shapes under the Schwinger
mechanism.

Acknowledgments

This work was supported by the National Natural Science Foundation of China
(NSFC) under Grant No. 12375240 and No. 11935008. The computation was carried
out at the HSCC of the Beijing Normal University.

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# Why and how to implement worked examples in upper division theoretical
physics

Philipp Scheiger<EMAIL_ADDRESS>Holger Cartarius  Research
Group Teaching Methodology in Physics and Astronomy, Friedrich-Schiller-
University Jena, 07743 Jena  Ronny Nawrodt  Physics Didactics Research, 5.
Physikalisches Institut and Center for Integrated Quantum Science and
Technology, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart 70569,
Germany

###### Abstract

Studying worked examples has been shown by extensive research to be an
effective method for learning to solve well-structured problems in physics and
mathematics. The effectiveness of learning with worked examples has been
demonstrated and documented in many research projects. In this work, we
propose a new four-step approach for teaching with worked examples that
includes writing explanations and finding and correcting errors. This teaching
method can even be implemented in courses in which homework performance
constitutes part of the grading system. This four-step approach is illustrated
in the context of Lagrangian mechanics, which is ideal for the application of
worked examples due to its universal approach to solve problems.

##  I Introduction

Worked examples are step-by-step solutions of exercises or tasks. When well-
structured worked examples are included in the learning process of novices,
they improve the initial acquisition of cognitive skills as compared to
conventional problem-solving activities (the so-called worked-example effect)
[1, 2, 3]. Effective learning with worked examples can consist, for example,
of studying several solutions and explaining them oneself before solving a
similar problem without additional support (example-problem pairs [1]).

The advantages of this method are often understood through cognitive load
theory [3]. This theory assumes that learning is associated with cognitive
load and describes what can facilitate or impede learning. For example, the
removal of extraneous tasks such as remembering how to complete mathematical
operations allows the student to concentrate on the main learning task. In
addition, worked examples are considered cognitive activating, meaning that
they cause the student to focus attention on the most important learning tasks
(for an overview, see Refs. [4, 5, 3]). Learning with worked examples can be
more effective than to conventional problem solving because when learners are
presented with a new topic, they often concentrate cognitive resources on the
technical aspects of solving the problem. In that case, less resources are
available for the construction of abstract schemata, which lead to transfer
and can help to solve related problems (cf. Ref. [6]). The effectiveness of
such a worked examples effect is well examined and proven in well-structured
areas of mathematics and physics [3, 7, 8, 9, 10, 11, 12], and also in text
comprehension [13], and essay writing [14].

However, worked examples are not always superior or more efficient in learning
in comparison to classical problem solving [15]. They need to be structured
and designed in a way that extraneous load is decreased and germane load is
increased such that learners profit from them. For example, self-explanations
by students turn out to be important [16]. The request for self-explanations
(and also finding or fixing errors [17, 18]) in our approach is an important
difference from the simple presentation of solved problems in lectures.

The added value of worked examples has also been demonstrated for double
integrals in calculus at university level [19], i.e., in a topic which is
needed regularly in theoretical physics. Therefore, worked examples are a
promising tool for improving teaching in upper-division theoretical physics.
They can be used as further methodical variation (in addition to, e.g., peer
instruction and small group tutorials) for the transformation of upper-
division physics courses [20, 21, 22, 23] or to help students to overcome
typical difficulties with mathematical tools [24, 25, 26].

In addition to example-problem pairs (the simplest and most often recommended
scheme of worked examples [1, 11]), other approaches with additional
intermediate tasks can be beneficial [27] such as including self-explanation
prompts and successively removing more and more worked-out tasks in solutions.
It is the purpose of this work to show that a four-step approach based on
worked examples can indeed be introduced in theoretical physics. In Sec. II we
present the concept and explain its introduction in exercise courses. Short
notes on our experiences from several courses are given in Sec. III, and
conclusions are drawn in Sec. IV. More details based on the relevant effects
from cognitive load theory (cf. Refs. [29, 3, 30, 31, 16, 32, 33, 34, 35, 1,
11, 17, 36, 27, 28]) relevant for the concept and our conclusions for worked
examples in theoretical physics can be found in the supplementary material
bellow this paper.

##  II Concept of a four-step approach applied in theoretical physics

The goal of the four-step approach is to strengthen the worked example effect.
It ensures that extended self-explanations are taken seriously by all students
due to two additional steps embedded between completely elaborated worked
examples and problem tasks without solutions. While the individual steps are
known and tested, we propose an arrangement ideally suited for courses that
assign problem sets. A lecturer can implement the method with very little
additional preparation time because it can be based on existing textbook
problems. It offers the students a broader variety of examples without
investing more time, and the opportunity to acquire a deeper understanding of
the underlying principles used in the exercises. The concept is structured as
follows.

The first step in our scheme is a maximally elaborated worked example with a
detailed step-by-step solution. The problem chosen for this step should be
paradigmatic for this type of physics task. To reduce extraneous load
generated by the necessary collection of information (split attention effect
[3]) the important aspects are highlighted in the problem description and the
structure of the solution is explicitly shown. Ideally the calculation path
and the explanation of the different solution tasks are close together on the
worksheet. The students’ mathematical capabilities should determine amount of
mathematical detail that is included.

After this preparation, we foster self-explanations in step two by demanding
written explanations from the students for a solution that is presented
without explanations. This example can be simpler than the first one to help
students get started. Since most learners are passive or superficial self-
explainers, [16] we recommend either an obligatory written explanation for
assignments that are submitted for grading or peer discussions for assignments
that are completed in tutorials.

In step three students are asked not only to explain the work, but also to
identify and correct errors that are purposefully included in the solution.
This task should be more challenging than step 2; therefore, the chosen
example should be more difficult again. We recommend 2-4 errors, such that the
students continue to search after finding the first error but the wrong
solution does not become too confusing. The focus should be on errors in the
translation of the physics problem into its mathematical description. One
calculation error should be included for a quick boost of accomplishment (see
supplementary material).

Step four consists of a problem without any solution and it is the student’s
task to develop and provide a full solution, which is the typical last step of
worked examples.

A detailed elaborate example for Lagrangian mechanics can be found in
supplementary material 2 (bellow this text), which fits in a 90-min exercise
class or can be completed by students at home. All examples are typical
textbook problems [37, 38] (an English version of the first book is also
available [39]) and can be adopted easily. In addition, we provide a
discussion about the lessons we have learned using worked examples, how they
can be embedded in instruction, and their applicability in other topics.

##  III Experiences with the four-step approach

We used this or a similar structure of worked examples in a set of different
courses from mathematics refresher courses to seminars accompanying classical
mechanics, electrodynamics, quantum theory, and thermodynamics/statistis
lectures. In all these setups we observed strong interest of the students and
their active participation. In each of our applications the majority of the
students was able to solve the last problem correctly. Thus, we found that
worked examples can be productive for several teaching situations in
theoretical physics. The example presented in supplementary material 2 (see
bellow this text) on Lagrangian mechanics was tested in a 90-min online
seminar with a very satisfying performance of the students. Some students were
able to complete the whole task within the seminar; a short homework
(completing the last example) remained for the others.

While our four-step approach is, in principle, very simple, we want to
emphasize the challenges. A great challenge is finding of the correct level of
difficulty for the learners. Novices with less prior knowledge need more
explanations, and error finding can overstrain them. However, redundant
information can reduce the learning effects for students with adequate prior
knowledge. Those two sides on the scale should be considered for every lecture
and course.

##  IV Conclusions

Due to our positive experiences, we recommend the application of worked
examples in theoretical physics and invite other lecturers to adopt and
enhance the approach of worked examples presented here in theoretical physics.

A nice side effect of worked examples with additional tasks (written self-
explanations and error finding) is that they can also be used to assess the
students’ performances. Suggestions on how this can be implemented can be
found in supplementary material 2.

Our presented scheme can be applied to every topic for which a universal
solution structure to problems exists and worked examples are usable in
general. Very typical examples are Lagrangian mechanics of the second kind and
solutions to the time-independent Schrödinger equation (see Table 1).

Table 1: Solution tasks in Langrangian mechanics of the second type and the time-independent Schrödinger equation |  Lagrangian mechanics |  |  Time-independent Schrödinger equation
---|---|---|---
Task 1 |  Set up the holonomic constrains and calculate the degrees of freedom $S$. |  |  Determine the dimension of the Hilbert space and set up the Hamiltonian according to the physics problem.
Task 2 |  Define generalized coordinates $q_{i}$ according to the holonomic constrains. |  |  Set up the eigenvalue equation.
Task 3 |  Set up the kinetic $T$ and potential $V$ energy. |  |  Determine the eigenvalues.
Task 4 |  Set up the Lagrangian. |  |  Determine the eigenvectors according to their eigenvalues.
Task 5 |  Determine the equations of motion for every generalized coordinate. |  |  Check for boundary conditions.
Task 6 |  Reduce the equations if possible. |  |

A systematic evaluation of the long-term performance of the students after
attending the worked-examples problem courses has to be relegated to future
research. This was not a topic in this work; however, an actual proof of the
superiority as compared to bare problem solving is desirable. Nevertheless,
previous research indicates a benefit of worked examples in university
environments [19].

## AUTHOR DECLARATIONS

The authors have no conflicts to disclose.

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###### Acknowledgements.

The authors are grateful for financial support by the Academy of Teaching
Support of Friedrich Schiller University Jena. This research was also funded
by the Bundesministerium für Bildung und Forschung (Federal Ministry of
Education and Research), at the Professional School of Education Stuttgart
Ludwigsburg in the project “Lehrerbildung PLUS,” grant No. 01JA1907A. This
project is part of the “Qualitätsoffensive Lehrerbildung,” a joint initiative
of the Federal Government and the Länder which aims to improve the quality of
teacher training. The authors also thank Thomas Rubitzko for stimulating
discussions at the beginning of this project.

##  Appendix A THEORETICAL FRAMEWORK

As highlighted in the main text, worked examples need to be well structured
and embedded in the learning process of novices to improve the initial skill
acquisition [1, 2, 3]. The superiority of learning with worked examples as
compared to conventional problem solving is often explained by the cognitive
load theory (e.g., [28, 4]). Cognitive load can be described as the amount of
working memory resources a person needs to fulfill a task. The cognitive load
theory describes three types of cognitive load that are relevant during
learning. There is the intrinsic load which refers to the complexity of the
learning contents in relation to a learner’s prior knowledge. The second type,
the extraneous load, refers to activities that are irrelevant (in general or
under certain circumstances) for learning. For example, the search for new or
forgotten information, such as the definition of a formula or a calculation
rule, generates extraneous load if it is not a particular requested learning
goal. In well designed worked examples this type of cognitive load is reduced
as much as possible. The third type, called germane load, refers to cognitive
resources that are bound by learning-relevant activities. Unlike intrinsic
cognitive load, which is generally considered invariant, teachers can also
manipulate germane load. Increased germane load aims not only to solve the
task (intrinsic load) but to maximize the learning effect in automating schema
and building understanding. Self-explanations of the step-by-step solution is
an example for such cognitive load. In contrast to extraneous load germane
load is desired and crucial when deep understanding of the learning contents
is requested.

The worked examples effect can be effective in learning because when learners
solve problems at the beginning of an unknown topic, significant cognitive
resources are invested in technical aspects the solution, rather than
dedicated to the construction of abstract schemata. It’s this construction
that leads to transfer and can help to solve related problems. Solving
problems at the beginning is usually accomplished by a means-ends-analysis
strategy. Here a substantial portion of cognitive load is needed to keep many
aspects in mind such as the current problem state, the goal state, differences
between them, etc. Remaining working memory capacities are therefore limited
and are no longer available for learning processes like the construction of
abstract schemata, which are crucial in theoretical physics. In contrast,
learners can concentrate on gaining understanding when studying worked
examples since they are freed from performance demands (cf. Ref. [6]).

Cognitive load theory explains why learning with worked examples is superior
to common problem solving in a broad variety of learning situations. So far
there few examples of applications in upper division physics. However, there
are no reasons why there should not be a positive effect. In particular,
examples from mathematics show that even complex tasks can successfully be
addressed with worked examples [19]. This renders them to a promising tool for
improving teaching in upper division theoretical physics. Thus, worked
examples should be useful in the transformation of upper-division physics
courses as it is pursued, e.g., in the Science Education Initiative (for an
overview see [20, 21, 22, 23]). In addition, worked examples might help to
support students to overcome typical difficulties with certain mathematical
tools (see, e.g. [24, 25, 26]) when they have to solve problems with long and
complex mathematical transformations.

###  A.1 DESINGS FOR WORKED EXAMPLES

Worked examples are not always superior or more efficient in learning as
compared to problem solving [15]. However when worked examples are structured
and designed in a way that extraneous load is decreased and germane load is
increased, learners do profit from them.

A good design to reduce extraneous load takes several effects of the cognitive
load theory into account (cf. [29, 3, 30]). These are the split-attention
effect, the redundancy effect and the expertise reversal effect. When learners
have to split their attention between at least two sources of information that
have been separated either spatially or temporally, extraneous cognitive load
is increased and the learning efficiency is decreased (split attention
effect). Information that is redundant or unnecessary has the same effect on
learners cognitive load (redundancy effect) or information that is already
known by learners and stored in their long-term memory (expertise reversal
effect). In order to motivate learners to study the examples and to process
the information in worked examples it is recommended to demand them to solve a
similar problem after studying the worked example.

Besides decreasing extraneous cognitive load we want to increase germane load.
Therefore we use the self-explanation effect and the studying error principle.
We want to foster self explanation because learners who study examples longer
and explain them more actively to themselves are more successful [31]. But
most learners are passive or superficial self-explainers according to Renkl
[16]. This leads to the conclusion that demanding self-explanations by
instructional procedures is crucial in learning from worked examples. Self-
explanation does not necessarily mean a talking-aloud procedure. There is
evidence that prompting written self-explanations foster learning outcomes
[32, 33]. The self-explanation effect can be increased by explaining correct
and incorrect solutions [34]; especially explaining why incorrect solutions
are wrong helps to avoid these errors later [35]. The task to correct their
own mistakes in mid-term exams can help students to increase their performance
in the final exam [18]. However there is only a positive effect in finding and
explaining errors in worked examples for learners with adequate prior
knowledge [17], so implementing errors in worked examples too early can
overwhelm weaker learners with little prior knowledge. To prohibit such an
overchallenge, weaker learners need additional support by explicitly marking
errors [17] or by expert explanations and feedback why certain steps in the
solution are correct or incorrect [36].

Next to the most recommended scheme of worked examples, viz. example-problem
pairs [1, 11] other approaches with more intermediate steps are possible. In
many tasks it can be beneficial to combine self-explanation prompts and fading
or successively removing more and more worked-out steps in solutions as
Atkinson et al. [27] have shown. They found that this combination even
positively influences the quality of example processing for far-transfer
tasks, thus a scenario in which we wish to apply worked examples.

###  A.2 CONCLUSIONS FOR THE DESIGN IN THEORETICAL PHYSICS

To be most efficient in learning scenarios, worked examples must be designed
according to the effects and principles described above. However the more
complex the problem is, the more complex it is to take account of all these
effects in the given solution. This relation leads to a special challenge in
theoretical physics since it requires a broad spectrum of previous knowledge
and skills (mathematically and physically). However, this does not mean that
worked examples are not applicable. Santosa [19] implied that worked examples
can increase the learning efficiency in complex tasks of calculus such as
higher-dimensional integrals, exactly the type of mathematics many physics
problems demand to get solved. This result provides a strong motivation to
extend this approach to exercises in theoretical physics.

In addition to physical principles (e.g. laws, boundary conditions, etc.),
mathematical challenges (such as differential equations, differential &
integral calculus, linear algebra and so on) also increase the difficulty of
physics tasks. When students are not well versed in mathematical techniques
they have to interrupt their thoughts on the intended problem with a search
for (mathematical) solution methods. A search that can lead to other textbooks
or lectures as sources of information creates extraneous load due to the
split-attention effect. Thus the split attention effect is most crucial in the
applications we have in mind to reduce extraneous cognitive load. Reducing the
spatial split attention effect in such complex exercises is challenging due to
the amount of information. Nevertheless it can be minimized to a certain
degree by highlighting important information, correlations or dependencies in
the problem. In addition offering a step-by-step solution of a worked example
should reduce the temporal split attention effect.

Since students of theoretical physics are not total novices in problem
solving, our conclusion is mainly to foster self-explanations to increase
germane cognitive load in the continuative examples. One possibility is to
request written explanations for certain or all steps in the solution from
students [33, 32]. Another possibility is to extend the examples by certain
comprehension questions students are requested to answer. When students have
shown adequate problem solving skills or a broad prior knowledge, finding and
correcting errors should improve the effects of self-explanation even more.

##  Appendix B FOUR-STEP APPROACH IN LAGRANGIAN MECHANICS

This supplement provides a paradigm of how worked examples could be
implemented in theoretical physics. Worked examples are useful in every topic
in which there is a universal solution structure. To illustrate this approach,
we have chosen the topic of Lagrangian mechanics of the second kind. The
solution structure behind various application examples is itself a prime
example of solution structures and therefore predestined to illustrate “worked
examples.”

  1. 1.

Define the holonomic constraints and calculate the degrees of freedom $S$.

  2. 2.

Define generalized coordinates $q_{i}$ according to the holonomic constrains.

  3. 3.

Write expressions for the kinetic $T$ and potential $V$ energy.

  4. 4.

Set up the Lagrangian.

  5. 5.

Determine the equations of motion for every generalized coordinate.

  6. 6.

Reduce the equations if possible.

In addition, in theoretical physics, Lagrangian mechanics is usually the first
exposure of university students to a new topic, which is more conceptual than
Newtonian mechanics, and thus additional support is appropriate here. The
examples used in this paradigm can be found in modified form in textbooks like
[37, 38]. An English version of the first book is also available [39].

The first step in our scheme is a fully worked example with a detailed step-
by-step solution. After this preparation we foster self-explanations in step
two by demanding written explanations from the students to accompany a
solution that is provided. In step three this is extended by finding and
fixing errors with an explanation. Step four consists of a problem without any
solution and it is the student’s task to develop and provide a full solution.

The whole program fits in a 90 minute presence exercise course or can be dealt
with by students at home. All examples are typical text book problems and can
be adopted easily.

###  B.1 Step 1: Maximally elaborate solution - Pendulum on springs

As mentioned in the main text, the first step in our scheme is a fully worked
example with a detailed step-by-step solution. The split attention effect is
reduced by highlighting the important aspects in the problem description.
Explanations are given right beside each task of the solution procedure.
Mathematical explanations should be given where necessary depending on the
mathematical capabilities of the learners.

In the context of Lagrangian mechanics, the pendulum on springs (see Fig. 1)
was chosen as first example since it contains many typical features of
textbook exercises in Lagrangian mechanics. Thus, the detailed solutions of
this problem offers the learners a broad overview. Students can learn how to
deal with two coupled masses and two types of potential energy. For novices in
Lagrangian mechanics we recommend a very detailed solution for example by
explicitly naming all dimensions and arguing via holonomic constraints the
irrelevance of the $z$ component. If the students are already familiar with
holonomic constraints this part is redundant and should be excluded from the
worked example.

Exercise: Set up the Lagrangian (or Lagrangian function) and determine the
equations of motion. Reduce them as much as possible.

A point mass $m_{1}$ is attached via two springs with the same spring constant
$k$ between two walls. The equilibrium point of the springs corresponds to the
center position of the point mass between those walls. The restoring force is
$|\mathbf{F}|=k|\mathbf{x}|$. Mass $m_{1}$ is only able to move horizontally
along the $x$-axis. A second point mass $m_{2}$ is attached to $m_{1}$ via a
massless rod with length $l$. The second point mass can oscillate in the
$x$-$y$-plane under influence of a homogeneous gravitational field with force
$\mathbf{F_{G}}=-m_{2}g\mathbf{e_{y}}$. The angle of deflection is given by
$\varphi$ (see Fig. 1). The small-angle approximation applies.

Table 2 shows the solution path alongside detailed explanations for each
calculation step.

Table 2: Worked example for step 1. Next to the solution steps are detailed
explanations to support the students.

Set up the holonomic constrains and calculate the degrees of freedom $S$.
---
$\\!\begin{aligned} z_{1}=0,\;&z_{2}=0\\\ y_{1}=0,\;&(x_{1}-x_{2})^{2}+y_{2}^{2}=l^{2}\end{aligned}$ |  The masses do not move in the $z_{1}$, $z_{2}$ and $y_{1}$ coordinate. The $x_{2}$, $y_{2}$ components of mass 2 depend on $x_{1}$ and the rod’s length $l$. The dependency is defined by the Pythagorean theorem.
$\\!\begin{aligned} S=6-4=2\end{aligned}$ |  Two free particles have six degrees of freedom (2 times the 3 spatial dimensions). Since there are four constraints, the total number of degrees of freedom reduces down to two.
Define generalized coordinates $q_{i}$ according to the holonomic constrains.
$\\!\begin{aligned} q_{1}=x_{1}=x,\qquad&q_{2}=\varphi\end{aligned}$ |  A wise choice for the generalized coordinates is the x position of mass 1 and the angle of deflection.
$\\!\begin{aligned} x_{1}=x,\qquad&x_{2}=x+l\sin\varphi\\\ y_{1}=0,\qquad&y_{2}=-l\cos\varphi\\\ z_{1}=0,\qquad&z_{2}=0\end{aligned}$ |  The coordinates are chosen such that mass $m_{1}$ is in the center in the equilibrium state. The transformation between Cartesian and the generalized coordinates is helpful for setting up the Lagrangian.
$\\!\begin{aligned} \dot{x}_{1}=\dot{x},\qquad&\dot{x}_{2}=\dot{x}+l\dot{\varphi}\cos\varphi\\\ \dot{y}_{1}=0,\qquad&\dot{y}_{2}=l\dot{\varphi}\sin\varphi\\\ \dot{z}_{1}=0,\qquad&\dot{z}_{2}=0\end{aligned}$ |  The derivative of time delivers the required velocity. Therefore the time dependency of all variables and the chain rule must be taken into account.

Set up the kinetic $T$ and potential $V$ energy.
---
$\\!\begin{aligned} T&=\frac{m_{1}}{2}\dot{x}^{2}_{1}+\frac{m_{2}}{2}\left(\dot{x}^{2}_{2}+\dot{y}^{2}_{2}\right)\\\ &=\frac{m_{1}}{2}\dot{x}^{2}+\frac{m_{2}}{2}(\dot{x}^{2}+l^{2}\dot{\varphi}^{2}+2l\dot{x}\dot{\varphi}\cos\varphi)\end{aligned}$ |  The overall kinetic energy is the sum of all velocity components. $y_{1}$, $z_{1}$ and $z_{2}$ contribute nothing since there is no motion in these directions.
$\\!\begin{aligned} V_{1}&=\frac{k}{2}(-x)^{2}+\frac{k}{2}x^{2}=kx^{2}\\\ V_{2}&=-m_{2}gl\cos\varphi\end{aligned}$ |  The potential energy of the first point mass is the energy stored in the two springs. The second mass swings in a homogeneous gravitational field.
Set up the Lagrangian.
$\\!\begin{aligned} L&=T-V\\\ &=\frac{m_{1}+m_{2}}{2}\dot{x}^{2}+\frac{m_{2}}{2}l^{2}\dot{\varphi}^{2}+m_{2}l(\dot{x}\dot{\varphi}+g)\cos\varphi-kx^{2}\end{aligned}$ |  The Lagrangian is the overall kinetic energy minus the potential energy.
Determine the equations of motion for every generalized coordinate.
$\\!\begin{aligned} &\frac{d}{dt}\frac{\partial L}{\partial\dot{x}}-\frac{\partial L}{\partial x}=0\\\ &=\frac{d}{dt}\left[(m_{1}+m_{2})\dot{x}+m_{2}l\dot{\varphi}\cos\varphi\right]+2kx\\\ &=(m_{1}+m_{2})\ddot{x}+m_{2}l\ddot{\varphi}\cos\varphi-m_{2}l\dot{\varphi}^{2}\sin\varphi+2kx\end{aligned}$ |  Via the Euler-Lagrange equations the equations of motion can be determined. To do so, the derivatives must be calculated in the correct order.
$\\!\begin{aligned} &\frac{d}{dt}\frac{\partial L}{\partial\dot{\varphi}}-\frac{\partial L}{\partial\varphi}=0\\\ &=\frac{d}{dt}\left[m_{2}l^{2}\dot{\varphi}+m_{2}l\dot{x}\cos\varphi\right]+m_{2}l(\dot{x}\dot{\varphi}+g)\sin\varphi\\\ &=m_{2}l\ddot{\varphi}+m_{2}\ddot{x}\cos\varphi+m_{2}g\sin\varphi\end{aligned}$ |
Reduce the equations if possible.
$\\!\begin{aligned} (m_{1}+m_{2})\ddot{x}+2kx&=m_{2}l(\dot{\varphi}^{2}\varphi-\ddot{\varphi})\\\ &\\\ l\ddot{\varphi}+g\varphi&=-\ddot{x}\end{aligned}$ |  The approximation of small angles ($\sin\varphi\approx\varphi$, $\cos\varphi\approx 1$) simplifies expansions of trigonometric functions and reduces the equations of motion.
These two differential equations are strongly coupled and as the term
$m_{2}l\dot{\varphi}^{2}$ indicates not even linearly coupled but
quadratically. It is not trivial to solve such equations, but it is also not
part of this exercise.

Figure 1: Sketch of the two masses of example 1. $m_{1}$ is clamped between
two springs and $m_{2}$ is a pendulum fixed to $m_{1}$ in a gravitational
field.

###  B.2 Step 2: Fostering self-explanations - Mass in a cone in a
gravitational field

In this example, students are encouraged to formulate their own explanations.
A simpler example is sufficient for this task since the self-explanation is in
the focus. This helps the students to get started.

In Lagrangian mechanics the problem “Mass in a cone in a gravitational field”
(see Fig. 2 was chosen as second example. The problem is not as difficult as
example 1, since there is only one mass and one potential energy.

Exercise: Set up the Lagrangian and determine the equations of motion. Reduce
them as much as possible.

Figure 2: Sketch of example 2. A pellet moves under influence of gravity on
the inner surface of a cone with an aperture angle $\Theta$.

On the inner face of an upwards opened cone is a pellet with mass $m$. The
cone has an aperture angle of $2\Theta$ and the pellet can move frictionless
on its inner surface. It is under the influence of the gravitational force
$\mathbf{F_{G}}=-mg\mathbf{e_{z}}$. The axis of the cone is identical with the
$z$-axis and its top is located in the origin, see Fig. 2.

Table 3 shows the mathematical solution.

Table 3: Mathematical solution for step 2. Next to the solution steps is free
space for students’ written self-explanations. Set up the holonomic constrains
and calculate the degrees of freedom $S$
---
$\\!\begin{aligned} \tan\Theta=\frac{\rho}{z}\quad\Leftrightarrow\quad z=\rho\cot\Theta\\\ S=3-1=2\end{aligned}$ |
Define generalized coordinates $q_{i}$ accordant to the holonomic constrains.
$\\!\begin{aligned} q_{1}&=\rho,\qquad q_{2}=\varphi\\\ x&=\rho\cos\varphi,\;y=\rho\sin\varphi,\;z=\rho\cot\Theta\end{aligned}$ |
Set up the kinetic $T$ and potential $V$ energy.
$\\!\begin{aligned} T&=\frac{m}{2}[\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}]\\\ &=\frac{m}{2}\left[(1+\cot^{2}\Theta)\dot{\rho}^{2}+\rho^{2}\dot{\varphi}^{2}\right]\\\ V&=mgz=mg\rho\cot\Theta\end{aligned}$ |
Set up the Lagrangian.
$\\!\begin{aligned} L&=T-V\\\ &=\frac{m}{2}\left[(1+\cot^{2}\Theta)\dot{\rho}^{2}+\rho^{2}\dot{\varphi}^{2}\right]\\\ &\qquad-mg\rho\cot\Theta\end{aligned}$ |
Determine the equations of motion for every generalized coordinate.
$\\!\begin{aligned} &\frac{d}{dt}\frac{\partial L}{\partial\dot{\rho}}-\frac{\partial L}{\partial\rho}=0\\\ &m(1+\cot^{2}\Theta)\ddot{\rho}-m(\rho\dot{\varphi}^{2}-g\cot\Theta)=0\end{aligned}$ |
$\\!\begin{aligned} &\frac{d}{dt}\frac{\partial L}{\partial\dot{\varphi}}-\frac{\partial L}{\partial\varphi}=0\\\ &m(\rho^{2}\ddot{\varphi}+2\rho\dot{\rho}\dot{\varphi})=0\end{aligned}$ |
Reduce the equations if possible.
$\\!\begin{aligned} (1+\cot^{2}\Theta)\ddot{\rho}-\rho\dot{\varphi}^{2}+g\cot\Theta=0\\\ \rho^{2}\ddot{\varphi}+2\rho\dot{\rho}\dot{\varphi}=0,\qquad\rho\neq 0\end{aligned}$ |

###  B.3 Step 3: Finding and fixing errors - Two masses on a wedge coupled by
a spring

To foster self-explanation the third example has errors intentionally
included. This task should be more challenging than step 2; therefore the
chosen example should be more difficult. In our example for Lagrangian
mechanics, an example with two generalized coordinates (Fig. 3) was chosen
such that the errors cannot be found easily. In the incorrect solution a
holonomic constraint is wrong. This error was chosen because students often
have problems with the transition between mathematics and physics. The second
error is located in the potential energy of the spring. Only the extension or
compression compered to the stress-free length is relevant for the potential
energy. The missing stress-free length in the potential is a typical student
mistake and is easily overlooked in a given solution.

Exercise: Set up the Lagrangian and determine the equations of motion of the
problem given in Fig. 3.

Figure 3: Sketch of example 3. Two masses on a wedge connected via a massless
spring in a gravitational field.

Two masses $m_{1}$ and $m_{2}$ move on a wedge. There is no friction but a
gravitational field given by the force
$\mathbf{F_{G}}=-m_{1/2}g\mathbf{e_{y}}$. The masses are connected via a
massless spring with a spring rate $k$ and a stress-free length $l$.

Set up the holonomic constrains and determine the number $S$ of degrees of
freedom

$\displaystyle z_{1}=0,\qquad
z_{2}=0,\qquad\frac{y_{1}}{x_{1}}=\tan\alpha,\qquad\frac{y_{2}}{x_{2}}=\tan\beta$
$S=6-4=2$

Define generalized coordinates, which conform to the holonomic constrains

$q_{1}=r_{1},\qquad q_{2}=r_{2}$

Transformations

$\displaystyle x_{1}$ $\displaystyle=-r_{1}\cos\alpha,\qquad$ $\displaystyle
x_{2}=r_{2}\cos\beta$ $\displaystyle y_{1}$
$\displaystyle=-r_{1}\sin\alpha,\qquad$ $\displaystyle y_{2}=-r_{2}\sin\beta$
$\displaystyle z_{1}$ $\displaystyle=0,\qquad\qquad\quad$ $\displaystyle
z_{2}=0$

Set up the kinetic $T$ and potential $V$ energy

$\displaystyle T$
$\displaystyle=\frac{m_{1}}{2}\dot{r}_{1}^{2}(\cos^{2}\alpha+\sin^{2}\alpha)+\frac{m_{2}}{2}\dot{r}_{2}^{2}(\cos^{2}\beta+\sin^{2}\beta)$
$\displaystyle=\frac{1}{2}(m_{1}\dot{r}_{1}^{2}+m_{2}\dot{r}_{2}^{2})$
$V=-m_{1}gr_{1}\sin\alpha-m_{2}gr_{2}\sin\beta+\frac{k}{2}(r_{1}+r_{2})^{2}$

Set up the Lagrangian

$\displaystyle L=T-V=$
$\displaystyle\frac{1}{2}(m_{1}\dot{r}_{1}^{2}+m_{2}\dot{r}_{2}^{2})+m_{1}gr_{1}\sin\alpha$
$\displaystyle\qquad+m_{2}gr_{2}\sin\beta-\frac{k}{2}(r_{1}+r_{2})^{2}$

Determine the equations of motion for every variable

$\displaystyle\frac{d}{dt}\frac{\partial
L}{\partial\dot{r}_{1}}-\frac{\partial L}{\partial r_{1}}=0$ $\displaystyle
m_{1}\ddot{r}_{1}-m_{1}g\sin\alpha+k(r_{1}+r_{2})=0$

$\displaystyle\frac{d}{dt}\frac{\partial
L}{\partial\dot{r}_{2}}-\frac{\partial L}{\partial r_{2}}=0$ $\displaystyle
m_{2}\ddot{r}_{2}-m_{2}g\sin\beta+k(r_{1}+r_{2})=0$

###  B.4 Step 4: Student’s task - Rotating mass on a tabletop connected to a
second hanging mass

Since learners take the self explanation more seriously, when they have to
solve a familiar problem by their own, the last step is the most important.
Here a suitable task contains again all typical elements but is not too easy
or is a transfer task. In Lagrangian mechanics we chose (Fig. 4) a rotating
mass on a tabletop connected to a second hanging mass as a fourth example
since many elements of the first three examples can be used in this solution.
The coupling of two masses can be found in examples 1 & 3, and the rotary
motion in example 2. However, the combination of both effects never appeared.
Thus, there is a transfer students have to master solving this problem.

Exercise:  Set up the Lagrangian and determine the equations of motion of the
following setup. Reduce them as much as possible.

Figure 4: Sketch of example 4. A rotating mass on a tabletop is pulled towards
an hole by a second mass in a gravitational field.

A mass $m$ rotates without friction on a tabletop. The mass is connected to a
second mass $M$ by a string of the length $l=r+s$. The second mass is below
the tabletop and is pulled downwards by the influence of the gravitational
force $\mathbf{F_{G}}=-Mg\mathbf{e_{z}}$.

###  B.5 Discussion

####  B.5.1 Lessons learned

As mentioned in the main text, we used this or a similar structure of worked
examples in two mathematics refresher courses prior to theoretical physics
lectures in classical mechanics and electrodynamics, in a further topic of
classical mechanics (Noether theorem), in electrodynamics (profiles for
Maxwell’s equations and derivation of the wave equation), in quantum theory
(verbalization of formulas) and thermodynamics/statistical physics (idealized
cycles and partition functions). In all these setups we observed active
participation of our students. The worked example approach was preferred by
students in initial test phases to the presentation of solutions by the
lecturer and was rated as more useful. The performance of our students in the
last problem was satisfying. We were also able to show our students more
problems in each topic this way. Thus, we experienced that worked examples can
be productive for several teaching situations in theoretical physics.

While the scheme presented here is in principle very simple, we want to
mention two aspects, which should be kept in mind. The first addresses the
prior knowledge of the learners. The great challenge for lecturers, who want
to use worked examples, is to adapt the correct level of difficulty for the
learners. Novices need more solution steps and explanations. Also error
finding can overstrain them. On the other hand redundant information can
reduce the learning effects for students with adequate prior knowledge. For
example, in previous tests of worked examples in Lagrangian mechanics of the
first kind we delivered only a mathematical solution without explanations.
Since our students were not total novices, we thought additional explanation
were redundant. But this approach overstrained them. Thus, our recommendation
is to deliver more rather than fewer explanations in the beginning.

For step 3 (finding and fixing errors) we recommend 2 to 4 errors, so that the
students continue to search after the first finding but the wrong solution
does not become too confusing. The type of errors should depend on the
learning objectives of the instructor. We typically focus on physically
incorrect terms but also miscalculations are possible to increase the number
of errors. In general we recommend errors in the translation of the physics
problem into its mathematical description, one calculation error for a quick
boost of accomplishment, and errors that are typical for the used kind of
problems. Here it can be helpful to skip the third step in one year and to
search in students solutions for patterns in their mistakes.

In addition, when existing problem sets are used for the worked examples, the
notion should be the same in every example, otherwise extraneous cognitive
load can be increased and the worked examples lose their efficiency. If these
two aspects are considered, we are convinced worked examples improve lectures
and exercises in theoretical physics as an upper division levels research in
mathematics [19] and our own experience implicate.

####  B.5.2 Embedding in instruction

A beneficial side effect of the self-explanation fostering variants are their
use for lecture certificates or ungraded semester performances. In some
physics study programs students have to solve a certain amount of problems
every week to obtain an admission to the exam. This practice is justified with
the training students get by solving problems as a preparation for the exam
and it is meant to be a protection for students not to take an exam il-
prepared. As mentioned above worked examples can be superior to problem
solving. This is why we suggest to exchange the often used problem solving
exercises by worked examples which demand self-explanations or finding and
fixing errors.

However, if the students do not solve all problems by themselves and a rating
of their work is required, the rating system has to be adapted. Instead of
taking just the number of correct solution steps or solved problems into
account, it is possible to establish a rating on the steps of the worked
example scheme presented above. The quality of the students explanations in
the self-explanation tasks as well as the number of detected errors in the
error finding problem can legitimate the lecture certificate. In particular,
the number of problems that can be utilized for the rating does not change
much. We have had good experience by exchanging two problems without
assistance with four worked examples of the types described above.

The scheme presented does not need to be rigidly adopted by other instructors.
Most important are the presentation of the solution structure (step 1) and the
student’s task (step 4). The steps in between can help to ease the transition
from understanding a solution to producing a solution yourself. Therefore,
this four-step approach can also be split or modified and adapted to time and
structural constraints. While the four-steps approach filled 90 minutes in our
case, the four steps can be split up due to time restrictions into consecutive
seminars (down to 45 minute seminars). The time constraints and the level of
the students must be weighed here. It is conceivable to outsource steps 1 and
2 in the homework or to omit one step. For teaching students we once skipped
step 2 in favor of a educational introduction of worked examples and cognitive
load theory.

####  B.5.3 Applicability of the structure in other topics

In every topic were there is a universal solution structure to problems
similar to that in Lagrangian mechanics our presented scheme and worked
examples in general are applicable. For example solving the time-independent
Schrödinger equation

$\operatorname{\hat{H}}|\Psi\rangle=E|\Psi\rangle$ (1)

with a solution structure like:

  1. 1.

Determine the dimension of the Hilbert space and set up the Hamiltonian
according to the physics problem.

  2. 2.

Set up the eigenvalue equation.

  3. 3.

Determine the eigenvalues.

  4. 4.

Determine the eigenvectors according to their eigenvalues.

  5. 5.

Check for boundary conditions.

Since physics problems occur in well structured domains, there are many topics
suitable for worked examples. Just to inspire the readers mind we want to
mention a few examples among many others. There are calculations with work
integrals, or inertia tensors, Noether’s theorem and Hamiltonian mechanics in
classical mechanics. In quantum physics besides the time-independent
Schrödinger equation there are perturbation theory and the determination of
Clebsch-Gordan coefficients. In classical electromagnetism many solution
methods of Poisson’s equation like the multipole expansion are suitable for
worked examples as well as are calculations for thermodynamic cycles or the
determination of partition functions in statistical mechanics.