Chapters 5, 6, 7 Concepts covered •  deBroglie ma8er wavelengths: rela;vis;c, and non-­‐rela;vis;c •  Introduc;on to the idea of wave func;ons, probability densi;es, superposi;on of waves, wave packets and envelopes •  The speed of a ma8er wave is not the same as the speed of the par;cle the wave describes •  Heisenberg uncertainty principle •  Classical standing waves, superposi;on of L and R traveling waves, quan;za;on arises due to boundary condi;ons •  QM standing waves are independent of ;me, superposi;on of L and R traveling waves, quan;za;on due to boundary condi;ons •  Schrodinger in 1D: NON-­‐rela;vis;c!, •  Ideas valid for all QM wavefunc;ons/Schrodinger equa;ons: probability density, normaliza;on constant, use separa;on of variables to get a spa;al only (or radial only, angular only) and ;me only Sch. eqn, •  Rules for wavefunc;ons that solve the Sch. Eqn: wavefunc;ons and first deriva;ves need to match at boundaries, not blow up at +-­‐ infinity •  Free par;cle wavefunc;on, no quan;za;on of energy •  Infinite poten;al well, quan;za;on of energy, poten;al, wavelength •  Expecta;on values are weighted averages of variables •  Operators are associated with each measurable in a QM system, an operator applied to the wave func;on produces the measurable •  Finite poten;al well (realis;c, not square), E > V0 and E < V0, in QM the par;cle can exist in classically forbidden regions •  Barriers: infinite and finite, reflec;on and transmission, tunneling into classically forbidden regions, E > V0, E < V0 •  Can draw wells from 0 to L, or from –L/2 to L/
2, only changes math, not final answer •  3-­‐D Schrodinger: now have 3 independent quantum numbers, degeneracy of energy levels b/c now have several ways to combine the quantum numbers to get the same value of energy •  3-­‐D Schrodinger in radial coordinates: can s;ll apply separa;on of variables to ;me independent, get 3 eqns –  For a central force poten;al (1/r), the two angular solu;ons won’t impact the E, and are valid for ANY central force problem –  Use lookup tables to find equa;ons for Radial and Angular variables (R, Y) •
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Quan;za;on of angular momentum, energy quantum numbers: n, l, m_l Selec;on rules for transi;ons in an atom H atom wavefunc;on: probability of finding par;cle in SPHERICAL SHELL (thus we ignore the angular dependence) Shells: energy and spa;al Total angular momentum and spin, introduces new quantum numbers (j, s, m_s) Spectroscopic nota;on Magne;c moments for spin and orbital, spliing of E levels, removal of degeneracy in presence of internal & external B field •  4 quantum numbers needed to EXACTLY specify the state of a par;cle: n, l, m_l, m_s (know s exactly from experiment, can create j once you know l and s) •  Pauli exclusion principle •  Fermions and Bosons •  Indis;nguishability of par;cles in QM •  Ground state of elements and periodic behavior •  Pertuba;ons required to cause transi;ons between states in atoms Chapters 5, 6, 7 Examples done in class •  Ch 5: finding debroglie wavelength of par;cles, comparing to compton, uncertainty on velocity due to uncertainty principle for a micro and macroscopic par;cle, uncertainty in frequency of a radar gun, applica;on of normaliza;on constraint to get the amplitude of the wavefunc;on, e+e-­‐ annihila;on, EM wave eqn sa;sfied by f(phi) •  Ch 6: solved free par;cle, infinite square well, finite poten;al well (not exactly, need to graph solu;ons to get full numerical answers), infinite barrier for E>V0, E<V0, finite barrier •  Ch 6: probability of finding an electron in certain x ranges for infinite well and n=5 state, infinite pot well with boundaries from –L/2 to +L/2 (vs 0 to L as done in lecture) •  Ch 6: not done in class but included with lecture notes – verifying explicitly that a given wavefunc;on is a solu;on to the schrod eqn, where is par;cle most likely to be found, expecta;on value, probability of something tunneling through an air gap •  Ch 7: (to be done in class on Thursday during review) L, radial solu;ons for H and expecta;on values, eigenfunc;ons for H evaluated at specific quantum numbers, spliing of E levels due to B field, combina;ons of quantum numbers Chapters 5, 6, 7 Suggested problems from book •  deBroglie wavelength of neutron, cosmic ray protons •  Ranges of frequency/frequency packets •  Normaliza;on of wave func;on •  Using wave func;on to find probability of par;cle’s posi;on •  Uncertainty principle applied to: par;cles created in radioac;ve decay, produc;on of classically disallowed par;cles in interac;ons, par;cles confined in the nucleus •  Energy levels of a par;cle confined in a box •  Par;cle in infinite square well: find energy levels, wavelength emi8ed in transi;on between levels, probability of it being at a certain posi;on/volume, nuclei, solu;ons for symmetric boundary of well •  Reflec;on and transmission of waves from poten;al barriers: infinite barrier, step poten;al •  Probability of finding par;cle in area of infinite step •  3-­‐D Schrodinger: energy level diagrams, ground state wavefunc;on •  Quan;za;on of angular momentum: possible values of l and m_l given n, total possible energy states •  Hydrogen atom (aka 3D spherical coordinates Schrodinger eqn): find the wavefunc;on, radial probability density •  Spin-­‐orbit effect and energy levels/degeneracy removal in an external field •  Total orbital momentum, spin, and angular momentum, quantum numbers •  Spectroscopic nota;on and iden;fying atoms •  Allowed transi;ons in atoms •  Difference in energy between states in atoms