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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

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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

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