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PHYS 342. Modern Physics 2. Course Elements. The Schrödinger Equation The Rutherford-Bohr Model of the Atom The Hydrogen Atom in Wave Mechanics Many-Electron Atoms Molecular Structure Statistical Physics Nuclear Structure and Radioactivity. 1. The Schrödinger Equation.
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PHYS 342 Modern Physics 2
Course Elements • The Schrödinger Equation • The Rutherford-Bohr Model of the Atom • The Hydrogen Atom in Wave Mechanics • Many-Electron Atoms • Molecular Structure • Statistical Physics • Nuclear Structure and Radioactivity
1. The Schrödinger Equation • Justification of the Schrödinger Equation • The Schrödinger Recipe • Probabilitiesand Normalization • Applications • The Simple Harmonic Oscillator • TimeDependence • Stepsand Barriers
2. The Rutherford-Bohr Model of the Atom • Basic Properties of Atoms • The Thomson Model • The Rutherford Nuclear Atom • Line Spectra • The Bohr Model • The Franck-Hertz Experiment
3. The Hydrogen Atom in Wave Mechanics • The Schrödinger Equation in Spherical Coordinates • The Hydrogen Atom Wave Functions • Radial Probability Densities • Angular Momentum and Probability Densities • Intrinsic Spin • Energy Levels and Spectroscopic Notation • The Zeeman Effect
4. Many-Electron Atoms • The Pauli Exclusion Principle • Electronic States in Many-Electron Atoms • The Periodic Table • Properties of the Elements • X-Rays • Optical Spectra • Lasers
5. Molecular Structure • The Hydrogen Molecule Ion • The H2 Molecule an the Covalent Bond • Other Covalent Bonding Molecules • Ionic Bonding • Molecular Vibrations • Molecular Rotations • Molecular Spectra
6. Statistical Physics • Statistical Analysis • Classical versus Quantum Statistics • The Distribution of Molecular Speeds • The Maxwell-Boltzmann Distribution • Quantum Statistics • Applications of Bose-Einstein Statistics • Application of Fermi-Dirac Statistics
7. Nuclear Structure and Radioactivity • Nuclear constituents • Nuclear Sizes and Shapes • Nuclear Masses and Binding Energies • The Nuclear Force • Radioactive Decay • Conservation Laws in Radioactive Decay • Alpha Decay • Gamma Decay • Natural Radioactivity
Justification of the Schrödinger Equation A fundamental equation required for describing the wave behavior of nonrelativistic particles. The conditions required for justification: • Energy conservation must be satisfied • The equation must be contestant with de Broglie hypothesis • The equation must be continuous (linear and single valued). The Schrödinger Equation
Like other waves, the free particle can be described by the equation And for time independent case, ) (1) Since (2)
=E One dimensional Time-independent Schrödinger equation. In which, U is the potential energy =U(x). For a free particle U(x)=0. E is the particle’s total energy. m is the particle’s mass, x is the position. Did this equation satisfy the conditions?
The Schrödinger Recipe • Using Schrödinger equation with discontinuous does not affect the continuity of . To describe the different regions of space we have to write different equations. • Solve Schrödinger equation to find for different situations. • By applying boundary conditions, several solutions can be obtained The Schrödinger Equation
Comparison between the use of Newton 2nd law of motion and Schrödingerequation
Probabilities and Normalization • The wave function describes the wave properties of the particle. • The squared absolute amplitude of gives the probability for finding the particle at a given location in space. • The probability density P(x), the probability per unit length in one dimension, is defined as The probability to find the particle in the interval dx which lies between x and x+dx. The probability to locate the particle varies smoothly and continuously. The Schrödinger Equation
The probability of finding the particle between x1 and x2 is the sum of all probabilities of finding the particle in the intervals dx, and is given by • Normalization: Since the total probability over the whole region is 1, =1 Which is called normalization condition.
Any solution to the Schrödinger equation, for which becomes infinite, must be eliminated. For example, if the solution is for the region x>0, then the first term must be eliminated as it leads to infinite value of when x approaches infinity. This is done by using A=0. In the region x<0, the 2nd term must be eliminated by using B=0.
Average value of x, xav Any physical quantity depending on the particle’s position will be determined with uncertainty as we are not uncertain about the particle's position itself. The probability of finding the particle at a particular position gives probable outcome of any single physical measurement or average value of a large number of measurements. If x1 is measured a certain number of times n1 and x2 is measured n2 of times, and so on.
Applications • The Free Particle F(x)=0 and so U(x)=constant, doesn’t change for all values of x. Let U(x)=0 and substitute in Schrödinger equation. The Schrödinger Equation
The solution of this equation is in the form which gives the energy values.
Particle in a Box (one dimension) Consider one dimensional box of length =L, in which a particle is moving freely, with boundaries, U(x)=0 at 0 ≤ x ≤ L, and U(x)=∞ at x<0, x>L This box is called infinite potential well. Ψ=0 outside and inside the box. A and B have to be found first.
At the boundaries, ψ must be continuous, therefore, at x=0 and at x<0 ψ =0 This yields B must be 0 Therefore the solution is limited to at x=L and x>L, ψ =0 Therefore, A sin kL=0
Or where n=1, 2, 3,…. In this way we can determine the values of energies the particle can have U=∞ U=0 U=∞ X=0 x X=L
The constant A is still undetermined, we can make use of the normalization condition for the whole region inside the box, =1→ =1 The solution is
Example 5.2 An electron is trapped in a one-dimensional region of length 1X10-10m. • how much energy must be supplied to excite the electron from the ground state to the first excited state? (111 eV) • In the ground state, what is the probability of finding the electron in the region from x=0.09X10-10 m to 0.11X10-10m. (0.38%) • in the first excited state, find the probability of finding the electron between x=0 and x=0.25X10-10 m. (0.25) Example 5.3
An electron is trapped in an infinitely deep potential well of width L = 106fm. Calculate the wavelength of photon emitted from the transition E4 → E3. (472 nm)
3.10 The state of a free particle is described by the following wave function ψ(x) = 0 for x < −3a = c for − 3a < x < a = 0 for x > a • Determine c using the normalization condition. (1/2√a) • Find the probability of finding the particle in the interval [0, a]. (1/4)
Example 5.3 • Show that the average value of x is L/2, independent of the quantum state.
The Simple Harmonic Oscillator The Schrödinger Equation
Time Dependence The Schrödinger Equation
Steps and Barriers The Schrödinger Equation
Basic Properties of Atoms The Rutherford-Bohr Model of the Atom
The Thomson Model The Rutherford-Bohr Model of the Atom
The Rutherford Nuclear Atom The Rutherford-Bohr Model of the Atom
Line Spectra The Rutherford-Bohr Model of the Atom
The Bohr Model The Rutherford-Bohr Model of the Atom
The Franck-Hertz Experiment The Rutherford-Bohr Model of the Atom
The Schrödinger Equation in Spherical Coordinates 3.The Hydrogen Atom in Wave Mechanics
The Hydrogen Atom Wave Functions 3.The Hydrogen Atom in Wave Mechanics
Radial Probability Densities 3.The Hydrogen Atom in Wave Mechanics
Angular Momentum and Probability Densities 3.The Hydrogen Atom in Wave Mechanics
Intrinsic Spin 3.The Hydrogen Atom in Wave Mechanics
Energy Levels and Spectroscopic Notation 3.The Hydrogen Atom in Wave Mechanics
The Zeeman Effect 3.The Hydrogen Atom in Wave Mechanics
The Pauli Exclusion Principle 4.Many-Electron Atoms
Electronic States in Many-Electron Atoms 4.Many-Electron Atoms
The Periodic Table 4.Many-Electron Atoms
Properties of the Elements 4.Many-Electron Atoms