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by: Jamaal McGlynn


Jamaal McGlynn
GPA 3.78


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Class Notes
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This 12 page Class Notes was uploaded by Jamaal McGlynn on Tuesday October 20, 2015. The Class Notes belongs to PHYS610A at San Diego State University taught by M.Bromley in Fall. Since its upload, it has received 23 views. For similar materials see /class/225319/phys610a-san-diego-state-university in Physics 2 at San Diego State University.




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Date Created: 10/20/15
Lecture 11 Never enuf time o Propagation Gaussian Wavepacket Section 51 0 Particle stuck in a box Section 52 Gaussian Wavepacket vs time o free particle stationary states are propagating waves went Aeikx t Be ikxt o BUT linear combo of separable solns is a T DSE soln l 00 mg Wm T27 W elm de M f2 00 we 0 o Gaussians are the special case to consider 7M1 W02 eiPOwh 6 962202 ze propagator 0 those formulae in terms of Unitary transforms we Ulttgtwltogtgt EgtE Ogte iEth 0 Where remember the apgt pgt U U IpgtltpIeiEthdp c this operator can be evaluated in the x basis 00 U 937157 56 lt93Ut95 gt ltxpgtltplx gt ip2t2mhdp OO 1 00 Z 2 m 2 1990 37 lh lp t2mhd zmx ac 2ht 27 we 6 p V Mine 0 which allows us to nd the wavefunction at any time lt93 tgt lt95Ut 0gt flt93Ut95 gtlt56 0gtd95 o in general xt fUxtx x Odx uncertainty principle 0 think of in terms of Gaussians minimum uncertainty 0 ie measure 9 get gaussian ie some uncertainty in p av at 1 was Let xt e h 2m 6 ma 39ht 0 0 so the Width of the moving wavepacket changes a 1 752152 ht t O h m 0 W204 ma p ma 0 so particle mass m measure a to nucleus 10 15m Av0 Upam O h 10 16msec m0 0 a bloody long time before macroscopic particle moves 09615 Planewave Phase Velocity 0 SO x t Aeikxwkt and k i 0 Where wk 73622771 wavelength A 27Tlc 0 Speed coef cient of If ever a coef cient wk vquantum k 2m hlk E 2m 2 0 but the classical speed is given by E mvclassy 2E velassy m 2vquantum Wavepacket Phase vs Group velocity 0 Plane wave w Asincx wt w 27TT k 27TA QM dispersion relation was Eh 73622771 Phase velocity Uphase wk his2m 0 Assume a wavepacket With peak was wo w6c 160 W t s we wet 0 0 ie wavepacket moves at vgmup wg dwdk hism All about dispersion relation 0 Note that deep water a g k gt Uphase 2119mmp o In shallow water w g d k gt Uphase vgmup httpgalileophysVirginiaeduolasses109NmorestuffAppletssinesGroupVelocityhtml The in nite square well 13 Lets solve SE with timeindependent potential Vx VOltxlta0 Vaoo Vx elsewhere lt7 r r r r r r r r r r r r r r r r r r r r 77gt 3 immediately implies that nx 0 when elsewhere inside the well we need to solve the free particle SE 752 d2 nx Which is re written as a simple harmonic oscillator eqn d2 n xZmEn dx2 h 72L nx where kn The in nite square well 23 The SHO has the general solution in terms of sin and cos nx An sincnx Bn coscnx To solve for coef cients impose boundary conditions Assume that nx is always continuous and that d nxdx is continuous except Where Vx oo 1M0 100 0 that s at the edges to Which we match at a O 10 An sinl nx Bn coslcnx Bn O and matching at a a 1mm An sincna 0 So either An O boring or sincna O kna O l7T 27T 37T The in nite square well 33 0 So the solutions to 1mm An sincna 0 kn 0 it i27Ta i37Ta 0 but kn O is also boring since nx O o and absorb Ve solutions sin cna sincna into A o Distinct solutions are kn mra with n 1 2 3 o Normalise foa An2 sin2l nx dx An2 1 2 mr 5262 M7 w sm a x 2m 277m2 n 1 WK woo woo n1 n2 3 x 39 x a a a in nite square well 4 3 w x w x w X o The normalisation A is independent of n SideNote A2 2a but we have Choice of phase of A Solutions alternate even and odd with n l nodes importantly solutions nx are mutually orthogonal in nite square well 53 c importantly solutions nx are mutually orthogonal 71 d1 3061 sin sin d9 otsam mm 1 Sin n7r sin 7m dx 7T m n mn 0 These wncr functions are also complete in that fx Eon wncr VEch sin 0 even if f has a nite number of nite discontinuities Stationary States 0 Which are Ilnx t nx gpnt nx 6 iEnth 2 n27r2 Ilnxt sin e 1 2ma2ht a 0 General soln to time dependent Schrodinger equation 00 2 TNT n27r2h x t Z on sin e 1 2ma2 t n21 a a 0 Expand initial wavefunction xnf O Z on on nxllx 0 dx g sin Q4930 dx 0 on tells you the amount of w in x t


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