YFX-1180
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QUANTUM MECHANICS YFX-1180 |
Description of some of the basic physical experiments underlying quant- um mechanics.Absolutely black body. Kirchhoff,Stefan-Boltzmann and Wien laws.Photoelectric effect.Compton effect. Bohr theory for Hydrogen atom.
The control questions for Lecture 1
Lecture 2
or take from original textbook (Pages:4-9).
Physical interpretation of the wave function. Probability Density. Normalizations of wavefunction and its physical meaning. Uniqueness, continuity and finiteness of the wave function. Schrödinger equation. Continuity equation. Probability current density. Stationary solution of the Schrödinger equation. Calculation of the values of measured phy- sical quantities as a solution to the corresponding eigenvalue problem. Operators in quantum mechanics. Eigenvalues and eigenvectors of the eigenvalue problem. A simple one-dimensional problem for a free parti- cle (calculation of eigenvalues and eigenvectors, discreteness of the solution for a periodic boundary condition, the corresponding eigenva- lue problem for the Hamilton operator and momentum operator, orthonor- mality of wave functions).
Orthonormality of wave functions for one-dimensional free periodic
motion, solution of the eigenvalue problem for the momentum operator.
Hermitian operators. Eigenvalues of Hermitian operators, ortho-
normalization of wave functions.
Completeness of wavefunctions of Hermitian operator. Representa-
tion of an arbitrary wave function as a linear combination of eigen-
functions of Hermitian operators, calculation of the series expansion
coefficients and their physical meaning, solution of the corresponding
eigenvalue problem.
Mean value of Hermitian operators.
Uncertainty principle.
The control questions for Lecture 3
Stepped barrier.
Propagation of particles over a stepped barrier.Cases E > Uo and E < Uo.
Reflection and transmission coefficients, classic limits.Calculations within the
framework of quantum mechanics. Energy dependence of reflection and transmission
coefficients. Penetration of particles into the barrier.
Rectangle barrier.
Tansmission coefficient in cases E > Uo and E < Uo. Resonants states and windows
of trasparency. Tunnel effect.
Infinite potential well. Finite potential well for stationary states E < Uo. Harmonic oscillator. Hermitian polynomial.Total energy. Normalizing of wave function. Creation and annihilation operators. Calculation of <x>, <Px> and <Epot>. Application for 1d lattice (Specific Heat calculation of a chain of atoms).
Angular momentum vector and operators. Square angular momentum operator. Presentation in spherical coordinates. Eigenvalue task for Lz and L^2 operators. Legendre polynomial. Spherical functions. Magnetic and orbital quantum numbers.
Hydrogen atom. Schrödinger equation. Total wavefunction (angular and radial parts). Energy of electron in hydrogen atom. Laquerra polynomial. Principal quantum number. Relations for principle, orbital and magnetic quantum numbers. Radial probability density. Rydberg constant. Laynman, Balmer and Paschen series. Bohr radius. Perio- dic table of elements. Aufbau principle.
Time independent perturbation theory (non degenerate case). Task formulation. Initial non perturbated task.General representation of the Hamilton operator, wave function and energy for the perturbed problem. First order perturbation theory equation. Calculation of the corrections for energy and wave function. Normalization of the perturbed wave function. Equation for second order approximation of perturbation theory. Calculation of the corrections for energy and wave function. Normalization of the perturbed wave function. Oscillator in constant external force field. Anharmonic oscillator for cubic and quadratic corrections of potential energy.
Time independent perturbation theory (degenerate case). Task formulation. Initial non perturbated task. General representation of the Hamilton operator, wave function and energy for the perturbed problem. First order perturbation theory equation. Secular equation. Calculation of the corrections for energy and wave functions. Stark effect.
Time dependent perturbation theory. Formulating of the problem for the original stationary unperturbed prob- lem (Schrödinger equation, non-stationary wave function). The representa- tion of wave function as a series of expantion. Formulating of the prob- lem for an external perturbation depending on time (Schrödinger equation, wave function). Equation for the time dependent expansion coefficients of the wave function.The relationship of these coefficients and the probabi- lity of the interlevel transition. Representation of the solution of this equation in the form of expansion into a series of perturbation theory approximations. Probability of transition for first order of perturbation theory approximation.The case of harmonic external perturbation.The "gol- den rule" of quantum mechancs. Relationship between the "golden rule" and spectroscopy.
Radiational transitions. Selection rules. Harmonic oscillator in a harmonically changing electric field. Selection ru- les for optical tansitions. Hydrogen atom in external electromagnetic wave. Transitions probability and selection rules.
Lecture 12 (link to last lecture)
Elements of A.Einstein radiation theory. Induced and spontanious
transitions.Probabilities of transitions. Physical reason for spontaneous
transitions.
Spin–statistics theorem.
The formal derivation of "classical" time-dependent Schrödinger equat-
ion and relation to classical total energy. The total energy of a particle
in the relativistic case and the transition to the corresponding relati-
vistic "Schrödinger" equation. Klein-Gordon equation. Dirac equations.Spi-
nor and Bispinor. Physical justification for the relationship between the
Pauli spin matrices and the spin magnetic moment of fermions.