Phyic 7 Graduate Quantum Mechanic Solution to inal Eam all 0 Each quetion i worth 5 point with point for each part marked eparately Some poibly ueful formula appear at the end of the tet In four dimenion there are i rotation operator ometime called L Ly L K Ky K Thee operator have commutation relation L L y K K y il Ly L Ky K il L L K K ily L K y K L y ik Ly K Ky L ik L K K L i Ky L K Ly K y L K0 (a) [9] Prove the three operator operator J L K are generalied angular momentum Thee are traightforward We imply compute each of the neceary commutator: J J y L K Ly K y L L y L K y K L y K K y i L K K L i L K ij Jy J Ly Ky L K Ly L Ly K Ky L Ky K il K K L il KiJ J J L K L K L L L K K L K K i L K K L i L K ij y y y y y y y Thee three commutation relation are ufficient to prove what wa requeted (b) [9] The three operator J L K are alo generalied angular momentum operator (you don t have to prove that) They all commute with J o Ji J j 0 Demontrate thi eplicitly for J ; ie how the three relation Ji J 0 We jut tart calculating: J J L K L K L L L K K L K K i L K K L 0 y y y y Jy J Ly Ky L K Ly L Ly K Ky L Ky K il K K L0 J J L K L K L LL KK LK K 0
(c) [7] Baed on what wa given or you have demontrated find four independent combination of rotation operator that commute You are allowed to quare them; for eample you could chooe J J a one of them if you thought it wa a good idea We know in general that for any angular momentum operator J commute with every component of J We can therefore pick the four operator J J J J The one without prime automatically commute with thoe with prime and J commute with J while J commute with J r where and r are two arbitrary poitive parameter and r i the ditance from the origin We wih to find eigentate of the reulting Hamiltonian with energy E (a) [6] Argue that eigentate wave function of the reulting Hamiltonian can be factored into a radial function and another function Decribe a completely a poible one of thee factor A pinle particle of ma in 3D i in the potential V r The potential i pherically ymmetric We know in general that the olution to uch problem are the product of a radial wave function R and an angular function Y and the angular function will be the pherical harmonic that i m The pherical harmonic Y function l m r R r Y l are completely known; it remain only to find the radial (b) [6] or the other factor write an ordinary differential equation that it atifie Copying from the equation given we have d ER rr l lr V rr rdr r d ER rr l lr R r R rdr r r (c) [3] Show that to leading order at mall r the radial wave function approimately k atifie the equation you found in part (b) if you aume R r r and olve for k a a function of l We ubtitute in the epreion everywhere but then want only to keep the lowet power of r o we have
There are three term that have d Er rr l l r r r r rdr r r k k k k k k k Er r l lr r r k k k k k k r and thee are all leading at mall r We therefore have k k 0 r k k k l l r r 0 k k l l We now olve thi quadratic equation for k namely k l l The negative root i actually unacceptable becaue it lead to a divergent value for the wave function 3 An electromagnetic field i decribed by the vector potential A m t e and calar potential U = 0 (a)[6] What are the electric and magnetic field for thi vector and calar potential? We calculate thee uing the formula A m EU ˆ t e A A y A A A y A BAˆ yˆ ˆ 0 y y ˆ (b) [6] Write Schrödinger time-dependent equation for an electron in thee field We can ue the Hamiltonian provided to find ge i H π eu BS π PeA t m m m m P m t Py P m i i m t t m y
(c) [7] Conider the wave function ep t m it it Show that im and m m We imply work it out: m P eai m tep it it im it m t m timm t im im im im im i im mm (d) [6] Show that the wave function in part (c) i a olution of Schrödinger equation It i obviou that 0 We therefore are attempting to check y i π t m m i ep it it t m m m i i iep it it mm m m m 5 A hydrogen atom i in the tate nl jm (a) [5] Which tate vector nlmm make up thi tate? j The value of n and l will till match However we now will have m and m varying but with three retriction: m mut be in the range - to m will only take on the value and ince J L S we mut have mj m m If we chooe m then mmj m So the two tate are 0 and
5 (b) [5] Write nl jm j a a linear combination of nlmm tate calling any unknown coefficient a b etc Write thee unknown coefficient a Clebch- Gordan coefficient (however for ubequent part of the problem call them a b etc) We write nl jm a 0 b j 5 5 The coefficient a and b are imply how you build up jm from the tate lmm ; ;0 and lmm ; ; o thee correpond to the C-G coefficient a ;0 b ; 5 5 r r (c) [8] Write the wave function r in term of the radial function and the pherical harmonic Ue any contant a b that you named in part (b) a needed The wave function for the tate vector nlmm i m Rnl r Yl m where m denote whether we are dealing the upper or lower component of r We therefore have r r a r 0 b r 5 0 0 ay ar ry br ry R r by (d) [7] or each of the following operator tell me what the reult of a meaurement of would yield Your anwer for each can be either a ingle anwer with a 5 probability of 00% or a lit of anwer with a correponding probability: L J J and L The original tate i an eigentate of L J and J and therefore there i no uncertainty in l l j j and the reult The reult will be the eigenvalue which are given by m repectively or L it i in a uperpoition of the tate nlmm 0 and nlmm The probabilitie are imply given by a for m = 0 and b for m = We therefore have in ummary: mm 5 which i
J L : P 6 J : P 35 : P L P a P b : 0 5 The double delta-well potential ha (at mot and for thi problem eactly) two bound tate and with energie E E or thi problem we will dicu what happen if we have two identical non-interacting particle in thi potential (a)[5] Two non-interacting ditinguihable particle of pin 0 are in thi potential Give a lit of all poible normalied bound quantum tate and their energie Energy State E We can put either particle in either tate or There E E are thu four poible tate a lited in the table at right The middle two are degenerate E (b) [8] Two inditinguihable particle of pin 0 are in thi potential Would thee be boon or fermion? Energy State Give a lit of all poible quantum tate and their E energie E E By the pin-tatitic theorem pin 0 particle are boon E which mean the tate mut be ymmetried The tate are lited at right Energy State E ; ; (c) [] Now conider inditinguihable particle of pin ½ in thi potential Keep ; ; in mind that in thi cae we mut include pin tate even when decribing ingle ; ; E E particle Give a complete lit of tate and ; ; their energie for two particle ; ; There are now four poible tate for a ingle particle and E ; ; To decribe the quantum tate we imply have to decide which two of thee four tate have particle in them not picking the ame tate twice ince thi would violate the Pauli ecluion principle We therefore have a total of i quantum tate which mut each be anti-ymmetried a hown above
6 In thi problem we will attempt to find the formula for the ermi preure for N identical non-interacting pin-/ fermion of ma m in a one dimenional bo of ie L We will ultimately work in the limit of large N and large L Some poibly helpful um are given below (a) [] or a pinle particle in one-dimenion the energy of the eigentate p i E p ml p How do the tate or energie change when we include pin? There i no change ecept the quantum tate mut include a pin deignation oon o they become m p p (b) [6] If we have N particle which tate will be occupied? You may aume that N i even What i the ermi energy E for N particle? Write your anwer both in term of N and L and in term of the one-dimenional particle denity n N L or each value of p there are two quantum tate If we aume that N i even then the tate p = through p N will be occupied The ermi energy i the energy of the highet filled tate or E pma N N ml ml 8mL 8m n (c) [0] ind the energy of all N particle Show that in the limit N (o N N ) it can be written in the form E NE and determine The energy of all N particle i the um of the energy for all the particle which i E E p N N N N N p p m ml 6 p ml ml N N N In the large N limit we approimate N N N o we have N E ml 3 3 NE (d) [7] Define the ermi preure a P E L n N L in the large N limit Show that it depend only on P The ermi preure (which in one dimenion i jut a force) i imply E N N N N N N N 3 n 3 3 L L ml ml ml m 3 M M M M 3 Sum: M p M M p M M M p M M 6 p p p p