Question

2h Question 20 marks Soalan 2 [10 markah) Harmonic oscillator problem can be solved based on operator and algebraic method where operators are given as a = (x + and a = (x 2) (a, and a is raising and lowering operator respectively) a) Prove that the commutator (a.a.) = 1. (Hints: Use commutator identities, lx.pl = ih and (x,x) = p.pl=0) [3 marks) b) Show that the harmonic oscillator Hamiltonian H+*x* can be written in the form H = hufa.a. + (Hints: find x and p in terms of the operator first) 17 marks)

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Answer 1

Oues) given mw mw a = mw (2+ ip 27 Q+ - Peising operator (x-il de-Lowering 2h operates we have to prove [at a ] 1 (ata.] = aaaa 0 mw Now, a da x - mio 27 p2 mtw ? mw ap tadi, 177 (22-22)] a. a. mw. x 2 + 2 25 mu? C. Z xp -px = lt ata e mw x2 + P² th 2h mtu? me Similarly Q_Az = mw/x+p? - 25 mw? mw ap-it 소 0.4+ = arep putting value from 0 to 0 (a.a.] = mw (x2+p2 x² pl mtoll mw (079.] =1 to mw 2h mwit Hence pscoved

given hamiltonism is H en nummula! atip Ix-il 2 x 24p2 mi w 2 mw xtil - ht 2-11 mw mw o 모 I mwly2 + 2 mw? The suite Themetoniom - [ 2 2m so that we may He appel (2412-10 By ep. 2 we com write. Im we (x² - t t. P2 944. = (Kw) (4mwe){24 tu (1 mwe) mo mos) Ata. = I (H-hw) mw 27 tw H = tw (ata +1) He Awfdplette Hamitoniam in tem of Atl.