Physics 235 Final Examination December 4, 2006 Solutions

Transcription

1 Physics 35 Fi Emitio Decembe, 6 Soutios.. Fist coside the two u quks. They e idetic spi ½ ptices, so the tot spi c be eithe o. The Pui Picipe equies tht the ove wvefuctio be echge tisymmetic. Sice the obit gu mometum is, the spti pt of the wvefuctio is echge symmetic. Thus the spi pt of the wvefuctio must be echge tisymmetic, i.e. the et spi of the uu system is (spis tipe). The d quk is so spi ½. Whe this is dded to the tot gu mometum of the uu system, the tot gu mometum of the poto, i.e. its spi, c oy be ½. b. If the two u quks hve etive gu mometum of, the spti wvefuctio is echge tisymmetic. The spi wvefuctio must the be echge symmetic, i.e. spi (pe spis). Addig spi to obit gu mometum, we get the possibe vues fo the tot gu mometum of the uu system of,, d. To get the possibe tot gu mometum of the ecited stte of the poto, we dd the d quk s spi ½ vectoiy to the uu gu mometum. The possibe spis e 5/, 3/, d /.. The poteti c be divided ito two pts, V( ) V ( ) + v( ) whee > V ( ) <. δ > v ( ) δ.e δ < The soutios fo V () e π cos,3,... eve pity ψ ( ) < π si,,... odd pity π E m Note tht the petubtio wi oy mi eve d odd pity soutios becuse v() is tisymmetic bout. () The petubed eegy eigevues e:

2 * + ψ ψ E E ( ) v( ) ( ) d δ π π E + cos d+ cos d o the sme itegs with si E sice the iteg is tisymmetic fuctio itegted symmeticy oud. Thus the eegies e t chged by the petubtio. (b) The petubed goud stte eigefuctio is: ψ whee ψ ( ) ψ v ( ) ψ E E π cos odd π v ( )cos d E π si eve The odd itegs e zeo fo the sme eso tht the eigevue iteg ws zeo. By usig the fct tht the eft hf of the eve iteg is the sme s the ight hf d chgig the vibe of itegtio to y π/, we get π δ si ycos ydy,,... E π δ cos( ) y cos( + ) y π E ( ) ( + ) δ + π E ( ) ( ) + δ π E δ π E ( ).,... π ( ) These tems dop pidy with, so tht oy the tem is sigifict. Thus ψ ( ) ψ ( ).8 ψ ( ) π /

3 3. We use the ti wve fuctio B ψ ( ) Ae whee B is the vibe costt. The vitio method estimte fo the goud stte eegy is the miimum of: E ψ H ψ d whee H + V( ) md The kietic eegy tem of Hψ is: d ψ d B BAe md md AB e B B B e m Thus we get the estimte of the goud stte eegy, usig the fct tht the iteg fom + is twice the iteg fom + : AB B AB B ( Bβ ) E mi e e + A e d m m AB y AB y A y mi e dy y e dy+ e dy 3 Bm m( B) B β (Note tht B must be gete th β/, o the st iteg i the fist ie bove is ifiity.) We get A fom the omiztio coditio: Thus * B A y B ψψd A e d e dy A B π B π B π B π E mi + m π π mb B β π mi B + m β B Settig the deivtive with espect to B equ to zeo gives: β 3/ m β B B ( ) β β B m B 3/ 3 βm β B B 3 β βm B B 3

4 Sove this fo B d iset it ito the epessio fo E bove.. As we sw i css, to good ppoimtio the dop i wve fuctio mpitude is just the dop i the size of the epoetiy fig tem iside the bie: A κ ( ') d' e m V [ E ] whee κ ( ) ( ) κ ( ) The tio of the mpitudes t the ed d begiig of the bie is: κ ( ) d κ () e κ ( ) The itesity tio, which is the tsmissio coefficiet, is κ( ) d κ( ) d κ () T e e κ ( ) sice the epoeti domites. I this pobem, V E κ ( ) mv( ) Sice hf of the poteti hs height V d the othe hf (-α)v, mv m( α) V κ ( ) d d + d mv m( α) V + mv ( α ) + mv α Thus the tsmissio coefficiet is α mv α α T e T T T whee T is the tsmissio coefficiet whe α. 5. The tsitio te is popotio to the sque of the mti eemet of the poteti betwee the iiti d the fi stte: * 3 V ψ ( ) V( ) ψ d b Fo eectic dipoe ditio, the poteti is popotio to the eectic dipoe momet, e. Both the iiti d fi wvefuctios e popotio to Y (, ) θ ϕ, which is costt. Thus the gu itegs e

5 π + dϕdcosθ Sice the d y compoets of e siθ cos ϕ d siθsiϕ espectivey, the zimuth iteg is i both cses. Tht eves the z iteg + + cos θ π cosθ d cosθ π Thus the mti eemet squed is, d the tsitio is fobidde. 5