Loeng 2 või võtke Loide raamatust (lk:4-9).

Teemad:

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).

Teise loengu kontrollküsimused

Loeng 3 (lk: 9-17)

Teemad:

    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.

Kolmanda loengu kontrollküsimused

Loeng 4 (lk: 18-26)

Teemad:

    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.

Mõned lisamaterjalid 4. loengu jaoks::

• PDF file: Propagation of particles over and under a rectangle barrier


• Image: Reflection and Transmission coefficients for stepped 1d barrier


• Javascript: Square of wavefunction for stepped 1d barrier and E < Uo
• Image: Square of wavefunction for stepped 1d barrier and E < Uo


• Javascript: Transmission coefficient for rectangle barrier
• Image: Transmission coefficient for rectangle barrier

Neljanda loengu kontrollküsimused

Loeng 5 (lk: 27-44)

Teemad:

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).

Some additional materials for 5-th lecture:

• Python code: Numerical calculation the energies of stationary states in finite well


• PDF file: Infinite potential well
• PDF file: Finite potential well
• PDF file: Harmonic oscillator
• Parabolic Cylinder Function
• PDF file: Application of a quantum harmonic oscillator to calculate the heat capacity
  of a one-dimensional chain of atoms.

The control questions for Lecture 5

Lecture 6 (Pages: 45-51)

Topics:

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.

The control questions for Lecture 6

Lecture 7 (Pages: 52-58)

Topics:

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.

Some additional materials for 7-th lecture:

• Periodic table of elements
• Visualization of atomic orbitals
• Aufbau principle I
• Aufbau princilple II
• Electron configuration I
• Electron configuration II


The control questions for Lecture 7

Lecture 8 (Pages: 102-107)

Topics:

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.

The control questions for Lecture 8

Lecture 9 (Pages: 107-114)

Topics:

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.

The control questions for Lecture 9

Lecture 10 (Pages: 115-120 )

Topics:

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.

Lecture 11 (Pages: 120-125 )

Topics:

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.

The control questions for Lecture 10-11

Lecture 12 (link to last lecture)

Topics:

    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.

The control questions for Lecture 12