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The Lorentz-Drude theory explains the motion of electrons in metals. It postulates that: 1) The outermost electrons of metal atoms are weakly bound and move freely inside the metal, behaving like gas molecules. 2) These "free electrons" are in thermal equilibrium and obey Maxwell-Boltzmann statistics. 3) When an external electric field is applied, the free electrons accelerate in the opposite direction but collide with the metal ions, preventing an increase in velocity. The electrons drift at a constant velocity.
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- Additive and multiplicative identities and inverses
- Conjugates, modulus, and triangle inequality
- Polar form representation using modulus and argument
- Exponential form for products, quotients, and powers
- Roots of complex numbers and finding nth roots
- Representing functions of a complex variable using modulus and argument
lec42.ppt
1) The document discusses examples of calculating the Jacobian of transformations. It defines the Jacobian as the determinant of the partial derivatives of the transformed coordinates.
2) It then discusses Möbius transformations, which are fractional linear transformations of the form (az+b)/(cz+d). The Jacobian of a Möbius transformation depends only on z.
3) Several examples are given of using Möbius transformations to map one geometric region to another, such as mapping a circle to a line.
lec41.ppt
This document discusses conformal mapping, which maps curves and regions in such a way that preserves angles and their directions. It provides examples of conformal mappings:
1) The mapping w = ez maps a vertical line in the z-plane to a circle in the w-plane, with the phase angle increasing along the circle.
2) The mapping ω = eiθ0(z-z0)/(z-z0) maps an area in the upper half z-plane to the interior of a unit circle in the ω-plane. Points on the x-axis in z are mapped to the boundary of the circle.
lec39.ppt
The document discusses Taylor and Laurent series expansions. It provides examples of using these expansions to represent functions around points.
Taylor series provides a power series representation of an analytic function around a point. Laurent series allows representing functions in annular regions, including points where the function is not analytic, using both positive and negative powers of (z - z0). Examples show deducing Laurent series expansions for simple functions like z4 and 1/z4 around various points, and evaluating coefficients via contour integrals and the residue theorem. The document also gives an example of using a contour integral to compute a Greens function in many-particle physics.
lec38.ppt
1) Jordan's lemma is used to convert real integrals over the infinite real axis into complex integrals over a contour enclosing the real axis in the complex plane.
2) Several examples are provided of using residues and Jordan's lemma to evaluate definite integrals over the real line or infinite intervals that involve functions with poles, including integrals of x^2, sin(x)/x, 1/(x^2+a^2)^2, and sin(x)/(x(x^2+a^2)).
3) The technique involves closing the contour with a semicircle at infinity where the integral over the semicircle goes to zero by Jordan's lemma, leaving the original integral equal to the residue theorem applied to the
lec37.ppt
1) The document discusses evaluating contour integrals using the residue theorem. It provides examples of calculating residues and evaluating integrals where the contour encloses poles.
2) The residue of a function f(z) at a pole z=a is the coefficient of the (z-a)^-1 term in the Laurent series expansion of f(z) about z=a.
3) According to the residue theorem, the value of a contour integral of a function along a closed loop is equal to 2πi times the sum of the residues of the function enclosed by the contour.
lec23.ppt
1) The document discusses representation of the Dirac delta function in cylindrical and spherical coordinate systems. It shows that δ(r - r') = δ(ρ - ρ')δ(φ - φ')δ(z - z')/ρ in cylindrical coordinates and δ(r - r') = δ(r - r')δ(θ - θ')δ(φ - φ')/r^2 in spherical coordinates.
2) It also derives the important relation ∇^2(1/r) = -4πδ(r) and shows its application to the Laplace equation for electrostatic potential.
3) The completeness of eigenfunctions of harmonic oscillators and Legend
lec21.ppt
1. The Dirac delta function is an important concept in quantum mechanics and electrodynamics that describes an impulse or large force acting over a very short time interval.
2. The key properties of the Dirac delta function are that it is equal to infinity at a single point and zero everywhere else, and that the integral of the function over its entire range is equal to one.
3. The Dirac delta function can be used to find the value of an arbitrary function f(x) at a specific point a, as the integral of f(x) multiplied by the Dirac delta function over all x is equal to f(a).
lec20.ppt
This document contains a series of tutorial problems related to matrices and linear algebra. Problem 1 asks to invert a 3x3 matrix. Problem 2 asks to write a vector as a linear combination of two other vectors. Problem 3 involves finding the inverse, trace, and determinant of related matrices. Problem 4 proves a property about powers of similar matrices. Problem 5 diagonalizes a 2x2 matrix and finds its eigenvalues and eigenvectors.
lec19.ppt
This document discusses finding the eigenvalues and eigenfunctions of a spin-1/2 particle pointing along an arbitrary direction. It shows that the eigenvalue equation reduces to a set of two linear, homogeneous equations. The eigenvalues are found to be ±1/2, and the corresponding eigenvectors are written in terms of the direction angles θ and Φ. As an example, it shows that for a spin oriented along the z-axis, the eigenvectors reduce to simple forms as expected for a spin-1/2 particle. It also introduces the Gauss elimination method for numerically solving systems of linear equations that arise in eigenvalue problems.
lec18.ppt
This document discusses solving a mass-spring system as an eigenvalue problem. It:
1) Sets up differential equations to model the displacements of two masses connected by springs.
2) Transforms the coupled differential equations into a matrix eigenvalue equation.
3) Solves the eigenvalue equation to obtain the frequencies of oscillation for the two masses.
4) Combines the eigenvectors with complex exponential functions to obtain general solutions for the displacements of each mass over time.
lec17.ppt
This document discusses linear transformations and matrices. It introduces how linear transformations on physical quantities are usually described by matrices, where a column vector u representing a physical quantity is transformed into another column vector Au by a transformation matrix A. As an example, it discusses orthogonal transformations, where the transformation matrix A is orthogonal. It proves that for an orthogonal transformation, the inner product of two vectors remains invariant. It also discusses properties of other types of matrices like Hermitian, skew-Hermitian and unitary matrices.
lec16.ppt
This document discusses properties of symmetric, skew-symmetric, and orthogonal matrices. It defines each type of matrix and provides examples. Key points include:
- Symmetric matrices have Aij = Aji for all i and j. Skew-symmetric matrices have Aij = -Aji. Orthogonal matrices satisfy AT = A-1.
- The eigenvalues of symmetric matrices are always real. The eigenvalues of skew-symmetric matrices are either zero or purely imaginary.
- Any real square matrix can be written as the sum of a symmetric matrix and skew-symmetric matrix.
lec30.ppt
1) The document discusses calculating the moment of inertia tensor for a cylinder with radius R and height H. It is shown that the only non-zero components of the inertia tensor are Ixx = (3MH + 4MR2)/12, Iyy = Ixx, and Izz = MR2/2.
2) Equations for velocity, acceleration, and the Christoffel symbols in an arbitrary coordinate system are presented. Expressions for calculating acceleration in cylindrical coordinates using the metric tensor and Christoffel symbols are given.
lec28.ppt
Tensors obey algebraic properties including addition, multiplication, contraction, and symmetrization. Addition of tensors combines their components. Multiplication of tensors combines their indices and ranks to form a new tensor. Contraction sets equal a covariant and contravariant index, reducing the tensor's rank. Symmetric tensors do not change sign under index interchange, while antisymmetric tensors change sign.
lec27.ppt
1) The document discusses tensors with multiple indices and the cross product of two vectors A and B. The components of the cross product vector C are given by Ai Bj - Aj Bi.
2) It describes how tensor components transform between coordinate systems using transformations of partial derivatives. The transformation property for cross products is derived.
3) Tensors are defined by their rank, with the number of covariant and contravariant indices specifying a tensor's rank. Vectors have a rank of 1. Examples calculate tensor components in different coordinate systems.
lec26.ppt
1) There are two types of vectors - contravariant vectors whose components transform according to Equation 1, and covariant vectors whose components transform according to Equation 2.
2) The dot product of two contravariant or two covariant vectors is not independent of the coordinate system.
3) The dot product of a contravariant and a covariant vector is independent of the coordinate system.
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lorentz Copy of we
- 3. -:Lorentz Drudge-:Lorentz Drudge theory-:theory-: to explain the motion ofto explain the motion of electrons in metal, Lorentz andelectrons in metal, Lorentz and Drudge made the flowingDrudge made the flowing postulatespostulates
- 4. (1) Free electrons-: the outermost(1) Free electrons-: the outermost electrons of the constituent atomselectrons of the constituent atoms of a metal are most weakly boundof a metal are most weakly bound with the atoms. Hence thesewith the atoms. Hence these electrons get separated form theirelectrons get separated form their atoms and move freely insideatoms and move freely inside electrons are called the freeelectrons are called the free electrons.electrons.
- 5. (2) Conduction electrons – there(2) Conduction electrons – there are large number of free electronsare large number of free electrons inside a metal and they behave likeinside a metal and they behave like the molecules of a gas enclosed inthe molecules of a gas enclosed in a vessel. Hence the electronsa vessel. Hence the electrons inside the metal constitute a gasinside the metal constitute a gas which is also called the freewhich is also called the free electrons gas the thermal andelectrons gas the thermal and electron .electron .
- 6. (3) Statistics – the free electrons in(3) Statistics – the free electrons in thermal equilibrium obey the maxthermal equilibrium obey the max well bolt z man N statistics eachwell bolt z man N statistics each electrons has three degrees ofelectrons has three degrees of freedom for the translational motionfreedom for the translational motion (4) mean energy per electrons –(4) mean energy per electrons – according to max well bolt Z man Naccording to max well bolt Z man N statistics mean energy of electronsstatistics mean energy of electrons per degree of freedom.per degree of freedom.
- 7. (5) Random velocity – inside the(5) Random velocity – inside the metal the free electrons movemetal the free electrons move randomly with a high speed andrandomly with a high speed and their depend on the temperature oftheir depend on the temperature of metal during motion when thesemetal during motion when these free electrons collide the positivefree electrons collide the positive ions of the metal their speed andions of the metal their speed and direction of motion change suchdirection of motion change such that the number of electrons.that the number of electrons.
- 8. (6) Collisions drift velocity and(6) Collisions drift velocity and mean free path – when the metal ismean free path – when the metal is kept in an external electric field thekept in an external electric field the free electrons get accelerated in afree electrons get accelerated in a directions opposite to the externaldirections opposite to the external electric field but these is noelectric field but these is no increase in their velocity due to theincrease in their velocity due to the metal again and againmetal again and again
- 9. (6) Collisions drift velocity and(6) Collisions drift velocity and mean free path – when the metal ismean free path – when the metal is kept in an external electric field thekept in an external electric field the free electrons get accelerated in afree electrons get accelerated in a directions opposite to the externaldirections opposite to the external electric field but these is noelectric field but these is no increase in their velocity due to theincrease in their velocity due to the metal again and againmetal again and again