Such equations contain the electric field and the magnetic induction; as a consequence, their solution must be calculated in accordance with Maxwell's equations. Such equations contain the electric field and the magnetic induction; as a consequence, their solution must be calculated in accordance with Maxwell's equations.
A Review of Analytical Mechanics and Electromagnetism
Analytical Mechanics
- Introduction
- Variational Calculus
- Lagrangian Function
- Force Deriving from a Potential Energy
- Electromagnetic Force
- Work
- Hamilton Principle—Synchronous Trajectories
- Generalized Coordinates
- Hamiltonian Function
- Hamilton Equations
- Time–Energy Conjugacy—Hamilton–Jacobi Equation
- Poisson Brackets
- Phase Space and State Space
- Complements
- Higher-Order Variational Calculus
- Lagrangian Invariance and Gauge Invariance
- Variational Calculus with Constraints
- An Interesting Example of Extremum Equation
- Constant-Energy Surfaces
If the Lagrangian function does not depend on one of the coordinates, e.g. qr, the latter is called cyclically ignorable. The rest of the calculation is the same as in section 1.2; the two relations (1.64) turn out to be equivalent to each other and.
Problems
As Eis dictated by the initial conditions, the phase point of a conservative system always belongs to the same constant-energy surface. For a system with one degree of freedom, the relation describing the constant-energy surface reduces to H(q,p)=E, which describes a curve in the q p-plane.
Coordinate Transformations and Invariance Properties
- Introduction
- Canonical Transformations
- An Application of the Canonical Transformation
- Separation—Hamilton’s Characteristic Function
- Phase Velocity
- Invariance Properties
- Time Reversal
- Translation of Time
- Translation of the Coordinates
- Rotation of the Coordinates
- Maupertuis Principle
- Spherical Coordinates—Angular Momentum
- Linear Motion
- Action-Angle Variables
- Complements
- Infinitesimal Canonical Transformations
- Constants of Motion
Let V = V(x1,x2,x3) be the potential energy and E=const the total energy, and let A and Bindicate two points in space (x1,x2,x3) that limit the trajectory of the particle. It follows that the ˙ components are angular quantities M=r∧p, written in Cartesian and spherical references.
Applications of the Concepts of Analytical Mechanics
Introduction
Particle in a Square Well
The motion continues at a constant speed until the particle reaches position xM where it reverses again, and so on. 2E/basket keeps this value until the particle reaches the other edge of the pit, +xM.
Linear Harmonic Oscillator
Considering first the case of 0 ≤ E ≤ V0, the motion inside the well is restricted and the velocity of the particle is constant in the interval −xM < x <+xM, where the Hamiltonian function smx˙2/2 =E liver. To treat the E > V0 case, assume that the particle is initially at a position x <−xM.
Central Motion
On the other hand, M=r∧mr, so the constant˙ angular quantity is determined by the initial conditions of the motion. Since it is normal to the plane defined by randr, the trajectory of the particle always lies on such a plane.
Two-Particle Collision
The conservation of P due to the invariance under coordinate translations yields Pb=Pa, whence R˙b = ˙Ra. Moreover, (3.16) shows that it is also u1b = u1a and u2b =u2a, namely, in reference B the asymptotic kinetic energy for each particle is conserved separately.
Energy Exchange in the Two-Particle Collision
The conservation relation for the kinetic energyT1b+T2b=T1coupled with (3.22) gives the kinetic energy of the particle of massm1 after the collision,. 3.23) Expressing T1a andT1a in (3.23) in terms of the corresponding velocities gives the modulus of the final velocity of the particle of mass m1 as a function of its initial velocity and of angle χ,. In reference O, the angle between the final velocity of the particle of mass m2, initially at rest, and the initial velocity of the other particle has been defined above as θ =(π−χ)/2.
Central Motion in the Two-Particle Interaction
The analysis cannot proceed unless the form of potential energy V is specified.
Coulomb Field
System of Particles near an Equilibrium Point
They show that the relation Fj k = mjX¨j k, which yields the dynamics of the jth particle along the kth axis, involves the positions of all particles in the system due to the coupling of the latter. Define the 3N-dimensional vectorR=(X11,. . . ,XN3) which describes the instantaneous position of the system in the configuration space, and let R0 be a position where the potential energyVa has a minimum, namely (∂Va/∂Xj k)R0 =0 for all j,k.
Diagonalization of the Hamiltonian Function
On the right-hand side of (3.47), the term in the central bracket is replaced by the second term of (3.46). Finally, the potential energy term of (3.42) is reformulated in terms of bashTC h= bTL b, which is the required diagonal form.
Periodic Potential Energy
Specifically, (3.62) shows that the mean momentum varies so that its "coarse-grained" variation with respect to time, P /˜ τ˜ , is the negative coarse-grained variation of the Hamiltonian function with respect to space , - H /g = −δH /g. On the other hand, (3.58) shows that the coarse-grained variation of the position with respect to time, g/τ˜, is the derivative of the Hamiltonian function with respect to the mean momentum.
Energy-Momentum Relation in a Periodic Potential EnergyEnergy
In conclusion, are useful when one is not interested in the details of the particle's motion within each spatial period, but wants to investigate on a larger scale how the perturbation affects the average properties of the motion.
Complements
- Comments on the Linear Harmonic Oscillator
- Degrees of Freedom and Coordinate Separation
- Comments on the Normal Coordinates
- Areal Velocity in the Central-Motion Problem
- Initial Conditions in the Central-Motion Problem
- The Coulomb Field in the Attractive Case
- Dynamic Relations of Special Relativity
- Collision of Relativistic Particles
- Energy Conservation in Charged-Particles’ Interaction
The initial conditions are the same as in Section 3.6: the asymptotic motion of the first particle before the collision is parallel to the thex axis, while the second particle is initially at rest. This phenomenon is not taken into account in the analysis performed in Section 3.8, where the total energy of the two-particle system is assumed constant.
Electromagnetism
- Introduction
- Extension of the Lagrangian Formalism
- Lagrangian Function for the Wave Equation
- Maxwell Equations
- Potentials and Gauge Transformations
- Lagrangian Density for the Maxwell Equations
- Helmholtz Equation
- Helmholtz Equation in a Finite Domain
- Solution of the Helmholtz Equation in an Infinite Domain
- Solution of the Wave Equation in an Infinite Domain
- Lorentz Force
- Complements
- Invariance of the Euler Equations
- Wave Equations for the E and B Fields
- Comments on the Boundary-Value Problem
Other derivatives Le(w,wx,wt,x,t) with respect to combinations of arguments that do not appear in the first two equations (4.14) are set to zero. For this purpose, we take the rotation of both sides of the second equation in (4.24).
Applications of the Concepts of Electromagnetism
Introduction
Potentials Generated by a Point-Like Charge
This procedure is particularly useful when the field source is a single point charge. Seeing that the argument iδ in (5.3) is a function often', to complete the calculation one must follow the procedure described in Sect.
Energy Continuity—Poynting Vector
Momentum Continuity
Modes of the Electromagnetic Field
Since the vector potential is defined within a finite volume and also has a finite modulus, we can expand it in Fourier series. Since such functions are linearly independent of each other, (5.24) vanishes only if the coefficients vanish, that is, ak·k=0.
Energy of the Electromagnetic Field in Terms of Modes
Integrating over V uses the integrals (C.121) to get so that the fraction of electromagnetic energy arising from E reads. Such components are related to the polarization of the electromagnetic field above the plane [9, section.
Momentum of the Electromagnetic Field in Terms of Modes
The modulus of the individual momentum is equal to the energyWkσ relating to the same degree of freedom divided by c. Each summation in (5.43) is constant in time, so the electromagnetic momentum is conserved; as noted in Sects this is due to the periodicity of the Poynting vector.
Modes of the Electromagnetic Field in an Infinite Domain
It can be seen from (5.43) that the moment of the electromagnetic field is the sum of individual moments, each of which is associated with one degree of freedom.
Eikonal Equation
The description of rays obtained by approximating the eiconal equation is called Geometrical Optics. It shows that the equation is of second order in the unknown function(s), where the point of the ray corresponds to the abscissa of the curve along the ray itself.
Fermat Principle
Complements
- Fields Generated by a Point-Like Charge
- Power Radiated by a Point-Like Charge
- Decay of Atoms According to the Classical Model
- Comments about the Field’s Expansion into Modes
- Finiteness of the Total Energy
- Analogies between Mechanics and Geometrical Optics
2 Recalling the discussion of Sect.5.11.2, the use of (5.72) implies that the position of the particle deviates slightly from the center of the spherical surface. As anticipated at the beginning of this section, the planetary model of the atom is not stable.
Introductory Concepts to Statistical and Quantum Mechanics
Classical Distribution Function and Transport Equation
- Introduction
- Distribution Function
- Statistical Equilibrium
- Maxwell-Boltzmann Distribution
- Boltzmann Transport Equation
- Complements
- Momentum and Angular Momentum at Equilibrium
- Averages Based on the Maxwell-Boltzmann Distribution
- Boltzmann’s H-Theorem
- Paradoxes — Kac-Ring Model
- Equilibrium Limit of the Boltzmann Transport Equation
If the number of particles is large, the description of the dynamics of each individual belonging to the system is basically impossible. From fµ =fµ(H) one derives gradrfµ =(dfµ/dH) gradrH and gradpfµ = (dfµ/dH) gradpH, so that the equilibrium limit of the Boltzmann transport equation is .
From Classical Mechanics to Quantum Mechanics
Introduction
130 7 From classical mechanics to quantum mechanics In the last part of the chapter, the meaning of the wave function is given: for this, an analysis of the measurement process is first carried out, which shows the necessity of describing the statistical distribution of the measured values. of dynamic quantities when microscopic particles are processed; the connection with similar situations with massive bodies is also analyzed in detail. The chapter concludes with an illustration of the probabilistic interpretation of the wave function.
Planetary Model of the Atom
The positions of the atoms are marked by the dots visible in the upper part of the figure. The shape of the potential energy thus obtained can qualitatively explain several features of crystals.
Experiments Contradicting the Classical Laws
The product of the two factors thus found gives the expression for the spectral energy density of the black body. The perimeter is the intersections with the figure plane of the spherical waves produced by the scattering.
Quantum Hypotheses
- Planck’s Solution of the Black-Body Problem
- Einstein’s Solution of the Photoelectric Effect
- Explanation of the Compton Effect
- Bohr’s Hypothesis
- De Broglie’s Hypothesis
Changes in the energy of an atom are the result of outer shell electrons exchanging energy with the electromagnetic field. A meaningful generalization is the conservative motion of a particle subject to a force arising from the potential energy V(r).
Heuristic Derivation of the Schrödinger Equation
A function of the form (7.37) is considered to be a wave function associated with the motion of a particle at constant energy E= ¯hω. From this point of view, the analogue of the field intensity is the squared modulus of the wave function.
Measurement
- Probabilities
- Massive Bodies
- Need of a Description of Probabilities
Experiments show that the results depend on the size of the body being measured. If the filtered state is Ai and the measurement of the dynamic variable is repeated, the result is Ai again. 7.52).
Born’s Interpretation of the Wave Function
As noted in the previous sections, the hallmark of experiments performed on microscopic objects is the statistical distribution of results; Thus, a theory that adopts the wave function as a basic tool must identify the relationship between the wave function and such a statistical distribution. 154 7 From Classical Mechanics to Quantum Mechanics is proportional to the probability that a measuring process finds the particle inside the volume τ at that time.19 Note that the function used in (7.55) is the square modulus of ψ, that is, as pointed out in Sect . .7.5, the field intensity counterpart in the optical analogy.
Complements .1 Core Electrons.1Core Electrons
In the second case ψ is not normalizable:21 a typical example is the wavefunction of a free particle, ψ=Aexp [i (k r−ωt)]; however, it is still possible to define a probability ratio. Consider a particle whose wavefunction differs from zero at timet within a given volumeτ, and assume that a process of measuring the particle's position is initiated opt and completed at a later timet′; let the outcome of the experiment be improved information about the location of the particle, namely that the wavefunction differs from zero in a smaller volumeτ′⊂τ.
Time-Independent Schrödinger Equation
- Introduction
- Properties of the Time-Independent Schrödinger Equation
- Schrödinger Equation for a Free Particle
- Schrödinger Equation for a Particle in a Box
- Lower Energy Bound in the Schrödinger Equation
- Norm of a Function—Scalar Product
- Adjoint Operators and Hermitean Operators
- Eigenvalues and Eigenfunctions of an Operator
- Eigenvalues of Hermitean Operators
- Gram–Schmidt Orthogonalization
- Completeness
- Parseval Theorem
- Hamiltonian Operator and Momentum Operator
- Complements
- Examples of Hermitean Operators
- A Collection of Operators’ Definitions and Properties
- Examples of Commuting Operators
- Momentum and Energy of a Free Particle
As discussed in Section 8.2.1, the eigenfunctions of the Schrödinger equation for a free particle, for a given The eigenfunctions of the momentum operator are the same as those of the Schrödinger equation for a free particle.
Time-Dependent Schrödinger Equation
- Introduction
- Superposition Principle
- Time-Dependent Schrödinger Equation
- Continuity Equation and Norm Conservation
- Hamiltonian Operator of a Charged Particle
- Approximate Form of the Wave Packet for a Free Particle
- Complements
- About the Units of the Wave Function
- An Application of the Semiclassical Approximation
- Polar Form of the Schrödinger Equation
Such requirements, namely the continuity of the wave function and its first leads in space, were introduced in Sect. When the wave function is of the monochromatic type (9.4), the time-dependent factors cancel each other in (9.13) to obtain the value
