Quantum Mechanics and Basic Spectroscopy
Quantum Mechanics and Basic Spectroscopy CHEM 163
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This 6 page Class Notes was uploaded by Kylie Smitham on Monday September 7, 2015. The Class Notes belongs to CHEM 163 at University of California - Santa Cruz taught by Staff in Fall. Since its upload, it has received 44 views. For similar materials see /class/182239/chem-163-university-of-california-santa-cruz in Chemistry and Biochemistry at University of California - Santa Cruz.
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Date Created: 09/07/15
MATHEMATICAL VOCABULARY FOR QUANTUM MECHANICS Below are some concepts terms and examples of mathematical vocabulary relevant to our study of quantum mechanics 1 DIFFERENTIAL EQUATIONS The equations which determine as best as one can the behavior of particles on a quantum mechanical basis are differential equations where the desired solution a function satis es relationships involving derivatives The fundamental differential equation of quantum a 39 n J is the tim J r J equation r12 62wxtU t ithxt 2 6x2 woe 6 know solution by simple integration boundary conditions substitution to prove solution OPERATORS The concept of an operator is meaningful only in the terms of a rule which defines the operator39s effect on a function operator function 1 gt function 2 remember function number 1 gt number 2 Example s i Differential operators d df x a m dx T T T operator function result 61 2 x 2x dxlm ii Multiplicative operator note xop E fc in the text s notation Mathematical Vocabulary for Quantum Mechanics xap x xfpc T T T operator function 1 function 2 result xap3x3 x3x3 3x4 3 PRODUCT OF OPERATORS Examples d df xdf 1 xopl opfm mam E g opxopkx gm df fx yea m Note from examples ii and iii the results depend on the order of the operators u d d 1 ff 4 EIGENVALUES OF AN OPERATOR note ADp E 121 in the text s notation Aop x a ffx for somegsl x T T operator consts x is an eigenfunction of the operator10p with eigenvalue a Example 612 for ADP andfx sin kx dx 0P SEPARATION OF VARIABLES IN HYDROGEN ATOMS 3quot 2 i H39er 64 1 a 64 a r2 sine aeksme 69 1 r2 sin2 9 E 2 6r N 2414 74 UVV EW WU 9 RV 9 and Y9 9 2 ur2 2 2 ur Substitute 410quot 9 4 and multiply by WW 9 h2RrYe lt1 1 a 251R 1 Harlem9 4 LT z ne 4sin9EK 69 J 1 gtFEEJJZJEALO we 4sin29 a 2 L113 39 Set the terms in dotted box which depend on rto the constant 5 and the remainder of 9 4 dependent terms to 43 1 M 6Rr 2M mm 6r 7E W 1 a sine me 4 1 62Y6 4 Y9 sinewk ae Ye 4sin29 64 43 Multiplying equation 7 by Y 9 4 we get the rigidrotor Schrodinger Equation 2 anewg lt10 6L92 Y9 4 Li sine 69 9 Sin2 9 ad Y9 4 e CHEMISTRY 163A HEURISTIC MOTIVATIONS FOR SCHRODINGER WAVE EQUATION Schrodinger time independent wave equation Start with timeindependent classical wave equation for standing wave section 31 McQ ux t wx cos Zn t This can be Wiitten as sum of two traveling waves ux t WW cos 2n vt cos 211 Vt 211x 7 7L 2cos ux t wx cos 21tvt wx cos cat has With 7wV E V Substituting ux t into CWE eq 31 McQ d2 032 0 dx2 V2 w 61qu 4112 w 37 quot Make DeBroglie connection using 7 ofparticle gt mv gt E p7 h 2 1 2 P E EWN U mU hz E 2 2m 1 2m 77E U HEURISTIC MOTIVATION FOR SCHRODINGER WAVE EQUATION Using this in time independent equation for w 2 2 d w 411 w 72 quotE U1quot 4 12 dzw mw E Ulw 2 2 3911 h d as we W1 m YUpEp seethisis dx time independent Schrodinger Eqn II Second heuristic iTime dependence for free no particle energy particle wave A general complex time dependent wave is 21 ilt 7v axt age A Get relationship between space and time derivatives viaE hv p7 h 62axt Zni2 2quotiquot 3x2 Gal 7 e 2 axt 7 using p7 h 4112 2 hz p 61M 2 SinceE f m for free particle 62axt 4112 6x2 7 2mEaxt 712 62ax t 2maxr 6x2 taking time derivative 2m39v ax t usingE hv Chem 163A COMPTON SCATTERING V W I y pf we WV VA ml Aq RESULT 7M A thin2 mc 2 STARTING RELATIONSHIPS relationships from Planck and De Broglie E hv 6 p7 h 2 mv conservation of energy hv hv T h h conservation of momentum ye direction I fcose chosd conservation of mom entum y directi39on 0 x sine mVsin 1 2 3 4 5
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