Related Experiment Video
Updated: Jun 23, 2026
06:42
Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
Recursive algorithm for arrays of generalized Bessel functions: Numerical access to Dirac-Volkov solutions.
Erik Lötstedt1, Ulrich D Jentschura
1Max-Planck-Institut für Kernphysik, Postfach 10 39 80, 69029 Heidelberg, Germany. erik.loetstedt@mpi-hd.mpg.de
Summary
This study presents a numerically stable method for evaluating generalized Bessel functions, crucial for laser-matter interaction research. The technique avoids initial value computation, simplifying complex quantum electrodynamic calculations.
Area of Science:
- Theoretical physics
- Quantum electrodynamics
- Laser-matter interaction
Background:
- Generalized Bessel functions arise in relativistic and nonrelativistic treatments of laser-matter interactions.
- Evaluating these functions is essential for calculating cross sections of quantum electrodynamic processes in laser fields.
- Existing methods often require computing initial values, which can be numerically unstable.
Purpose of the Study:
- To develop a numerically stable method for evaluating generalized Bessel functions.
- To simplify the computation of quantum electrodynamic processes in laser fields.
- To demonstrate the utility of the method in exploring quantum-classical correspondence.
Main Methods:
- Utilizing a recurrence relation and a normalization condition for function evaluation.
- Avoiding the computation of any initial values for the generalized Bessel functions.
- Applying the method to Dirac-Volkov solutions for numerical calculations.
Main Results:
- A numerically stable method for evaluating generalized Bessel functions of arbitrary index and fixed arguments was demonstrated.
- The method successfully bypasses the need for initial value computation.
- Numerical calculations illustrated the quantum-classical correspondence of Dirac-Volkov solutions.
Conclusions:
- The proposed method offers a robust and efficient way to compute generalized Bessel functions.
- This advancement facilitates more accurate and stable calculations in quantum electrodynamics and laser-matter interaction.
- The method provides insights into the quantum-classical correspondence in relativistic quantum mechanics.
Related Concept Videos
01:20Bessel Function of Order Zero
A common physical example of wave propagation with radial symmetry is the ripple formed when a stone is dropped into a still pond. The disturbance originates at a central point and travels outward as a circular wave. As the radius of the wavefront increases, the same initial energy is distributed along a progressively larger circumference. Consequently, the amplitude, or height, of the wave decreases with distance from the center. This decay behavior cannot be captured by simple sine or cosine...
01:08Vector Algebra: Method of Components
It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
In many applications, the magnitudes and directions of...
In many applications, the magnitudes and directions of...
01:20Vector Calculus: Problem Solving
Vector calculus provides mathematical tools for analyzing physical fields that vary throughout space. One important application is the study of gravitational interactions between celestial bodies. Consider the Earth positioned at the origin and a satellite located at a point in three-dimensional space. The Earth exerts a gravitational force on the satellite, and this force can be described by components acting along the coordinate directions. Together, these components form a vector field that...
01:23Integrals of Vector Functions
Vector-valued functions provide a convenient framework for describing motion in space when both magnitude and direction are important. A drone’s velocity at any instant has a direction and a speed, and as the drone moves, both can change. A vector-valued function captures this behavior by assigning to each time a vector whose components are real-valued functions. Each component represents motion along a particular axis in space. Such functions can describe motion in either two-dimensional or...
01:22Second Derivatives and Laplace Operator
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
Consider a scalar function. The curl of its...
01:17Derivatives of Vector Functions
A vector-valued function describes position as a function of time. For example, in Cartesian coordinates, the position of a car moving along a curved road can be written as\begin{equation*}\textbf{r}(t)=\langle x(t),y(t),z(t)\rangle\end{equation*}Secant Vector and Average Velocity:This secant vector captures the overall change in position during the interval and provides a crude estimate of the direction of motion.At time t, the car is at point P, with position r(t). After a short interval h,...