Office: JILA A709; Phone ;

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1 Office: JILA A709; Phoe ; Problem Set 5 To be retured before the ed of class o Wedesday, September 3, 015 (give to me i perso or slide uder office door). 1. Ehrefest s Theorem: Ehrefest s Theorem states that quatum mechaical averages behave like classical properties, i.e., they obey classical mechaics. Check the validity of Ehrefest s Theorem for the case of the mometum of a particle i a geeral time-idepedet potetial V(x) by lookig at the time depedece of the average value of the mometum p (treat the oe-dimesioal case oly). Remember that the chage of mometum i time is give by the force i classical mechaics: p/t = F.. Particle i a box as a model for cojugated molecules: A crude treatmet of the electros i cojugated molecules regards these electros as movig i a 1-dimesioal box. Due to the Pauli exclusio priciple, o more tha two electros ca occupy the same level i the box (we will get the details later i class, they are ot ecessary for this problem). a.) For butadiee CH=CHCH=CH, which has four electros, the experimetal value for the wavelegth of the trasitio betwee the highest occupied level ad the lowest uoccupied level i the box is 17 m. Calculate the legth of the box. b.) What would the wavelegth of the trasitio with the lowest eergy be if the molecule loses oe electro from the system (i.e., make a catio)? ( credits) 1

2 Office: JILA A709; Phoe ; 6. Expasio of a wave fuctio: The eigefuctios x ( x) cos (for odd ) a a x ad ( x) si (for eve ) a a form a basis set i the state space for the particle i a box with the potetial V(x) = 0 for x a/ V(x) = for x > a/ We have see i class that we ca expad a wave fuctio as ( x) c ( x) a.) Calculate the coefficiets c for the triagular wave fuctio 3 x ( x) 1 for x a/ a a (x) = 0 elsewhere. You ca use a itegral table. b.) Plot the wave fuctio (x) ad the expasio i terms of the, trucatig the series after = 1, = 3, = 5, = 9. Set a = 1. c.) Calculate the average eergy of the system with the triagular wave fuctio (x) i uits of the groud state eergy E1 = h 8ma after = 3, = 5, = 9., trucatig the series

3 Office: JILA A709; Phoe ; 4. Potetial Steps a.) Cosider the situatio of a particle movig towards a step i the potetial eergy fuctio at x=0 as show below, where V(x) = 0 for x <0 ad V(x) = V0 for x 0. V(x) V 0 I II x=0 x Show that a plae wave with wave umber k1 i regio I is fully reflected at a step if E < V0, by calculatig the reflectio coefficiet. b.) Cosider a electro propagatig freely through a thi wire with a kietic eergy of 1 ev. The electro arrives at the ed of the wire, ad the work fuctio of the metal (i.e., the eergy ecessary to remove a electro from the metal) is 4 ev. Assumig that you ca treat this as a oe-dimesioal problem, how far will the evaescet wave fuctio of the electro peetrate ito vacuum? ( credits) 5. Operators a.) Cosider the operator Aˆ Bˆ zcˆ, where the operators Bˆ ad Ĉ are Hermitia operators, ad z is a complex umber. Determie  usig the defiitio of the adjoit operator. b.) The fuctios f ad g are both eigefuctios of the operator Ô, ad they are degeerate. Prove whether or ot the fuctio h = a f + b g (a, b complex) is also a eigefuctio of Ô. 3

4 Office: JILA A709; Phoe ; 6. The strict versio of HUP (Part 1): HUP i its strict versio states that xp ħ/ ad that this is a strict lower limit. Let s prove that. a.) First, let s defie the root-mea-square ucertaity of a observable A through its operator  as Show that this is the same as Aˆ A ˆ A. A Aˆ Aˆ. b.) Cosider the ket f xˆ ipˆ g, where is a arbitrary real umber, ad g is a ormalized wave fuctio. Calculate the square of the orm of f, which results i a secod order polyomial i. c.) Now, remember that the square of the orm of ay ket must be 0. Use this to derive a coditio for the discrimiat of the polyomial, leadig to d.) Now, defie two ew operators, Xˆ xˆ xˆ xˆ g xˆ g p ˆ xˆ 4 ad Pˆ pˆ pˆ pˆ g pˆ g, ad calculate [ X ˆ, Pˆ ]. ( credits) e.) Use the result of parts (b) (d) ad the defiitio of A from part (a) to obtai x ad p from Xˆ ad Pˆ ad thereby prove that is true. x p ħ/ ( credits) 4

5 Office: JILA A709; Phoe ; 7. A few easy pieces (1 credit each, o explaatio ecessary) a.) Ca wave fuctios that are ot eigefuctios of the Hamiltoia fulfill the timedepedet Schrödiger equatio (yes/o)? b.) For a particle i a potetial V(x), is the average mometum geerally coserved (yes/o)? c.) What is the form of the kietic eergy operator i mometum-space (there must be o operator symbols i the fial expressio)? 5