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Unformatted text preview: HIDTEZM SoLqTiou ‘4 ET We start with an n-dimensional Hilbert space spanned by vectors M + for 2' m 0, - - - ,n and I take these to be orthonormal as (A+i|)\+j) :61} (if I hadn’t started with such a basis I always could have constructed one using the Gram-Schmidt procedure). We know that these are eigenstates of the A operator as AlA—l—i) = (A+i)lA+i) (1) Now we deﬁne some operator B so that lRM=aB m where 0 5 a S n is integer. Let’s ﬁgure out how B acts on the basis states. We’ll call the state that you get when you act on a basis state with B by the name Mn) as BlA ‘l' ’3 W1) and then try to ﬁgure out what lwi) is in terms of basis states. To do this take [B, A] M + = aBlA + All/1i) (A + i _ a) la) Comparing to (1) it’s clear that BI/\+z') = W2.) is proportional‘ to |/\+ i ~a) so we write BIA-H) =c,-|/\+z'—a) (3) Z for some constants ci. (We don’t have enough information ot compute Ci.) Equation (3) tells us that B acts as a lowering operator. What about the Hamiltonian? We’re told that Hza—m 1A lot of people concluded that BIA +1) 2: [A +i - a), which is overly restrictive. There is no reason, in general, for B to return a properly normalized state. In particular, the action of some well known lowering operators like a and L- does not return a normalized state. l'Notz vomi- K mmnakbcmrvibwu, aWMCé will YW Lon/AW 1% in WC» git/ow HUB“? grace. M6 “0% on; ma Nob Vimafdfz: 4 an1l m acédw. ‘A+a‘——d> with Q = (3". Recalling that H = Hl we can write B+B" 2 then we can compute H in this basis since we know how B acts on the states. The matrix elements are H: Hi3» = (A+i|H|)\+j) Clearly EH 2 (A +i|H|A+j> = cj(A+i|A+j—a) = Cj5i,j—a and (Ble = B};- so 1 air Hij = ‘ lCi+a5iyj~a + Ciéi‘aijl 2 All this is for a 3i 0. When a = 0 then A, B commute so the basis states are simultaneous eigenvectors of both operators. Then the Hamiltonian will lie diagonal in this basis. :: X — N’ (YH’ .x a #3!» M haw, -.-. A” < N " -— . {a A (o rad __ .. . I. vac .‘ r ,. g-“rv --—:u I-o ' _ 'nL on .. V0444“ m-a’ ‘ .- a ' .. u‘ who , g A . , _ '«H um M . 3 ~ ((PA- £93 A=A+ A' a x r v A 1 co A w s A «N «v A, r. (a ‘V «WV a .. . J 4 ' /\-—— ., sa 5 “ma /\ = owe é‘Dr-o ‘vhw ’a N v..- n .z ; A '. ‘ u r. A ‘ aw. _ ____ﬂ ‘ ‘ v ' Wu. ‘ d¢maa+ H3 w ' “A d. -_. W. -LWLLJ v “6% w-“ :30 ; 004432 1’ ﬁlm =A. .. _______ \$35.! Aﬁng ngso _, < [MM—m; éﬁﬁléﬁl‘VgZ "‘ (KIIQALQ :_ @’412 4‘4-‘6 H’s) :0 (.1515 : ob<~b\ A www*._w_.___w . _ IL: A I. Co a * w :n z: _, l I an; ~ + +J. ' w ,ﬂmwQ. __,.,_, ft... : ,_ t+.L +1.4» —- u; .._I,. .. ' o 77 m. Tr 31L :— e o i¢~rﬁ . ‘ :3 A 9!” r g WQJP I w 46‘ \—~ och N . - 3: ,, mﬂﬂ MW M 2‘. m. m 3 3 r 5‘ . ‘ : ___, 1" r . v - “£15,-” ,“SILW4‘ x 4- ,957 A“- _-_r._rL.s1ﬂ§ ,m“ ##M We can write the hamﬂtonian and perturbation in matrix representation. E0 0 0 H92 0 Eu 0 0 0 Eu 0 —a D W: -a 0 ——a 0 —a 0 Thetotalhamﬂtonimiathen a) -a O H=Ho+Wn --6 En —-a 0 —a 8‘9 Note that the eigenvalues of Ho are all Bo and the eigenvecton are: (1) MP 0 O '0 Wm 0 0 Mo) 1 Initially, the system is in the state IMO» :2 Ml). Now, we can compute the eigenvalues and eigenvectora of H. 3133‘) 1 1 Wk) " 7; 01 Engo+af§ ->-(éi) Eaaﬁ'o-ax/i M» We can write the initial state in terms of the new eigenvectora. W0» = —‘— w - 1w + 11%) ﬂ 2 2 Then, we can apply the evolution operator. Mt» = Um W0» = w — Na) + 3%?“ w» ‘ z | If .. 33' ~ ._ “i . . . ‘ “W U i K. 4 -' . 5! Mb U I h - * v- :— AI<W v- * v- > t . w - an ‘._. ‘ ‘ u “ ~ wt ,_ c ' = M '. .... n e o ‘ 4" I ’ 41. ’4 1- 1 1 "‘l___L___—__._ . 3- A ____.____.—_..————-——- ’ A u; & - A ‘1 - H’Al1.44 I . ‘ '.. ‘ ‘ o , ,________.__._—— I - . . - m ‘5. H u 1- H m (Mun) = 04.] 4— z 6" (ml (MM) + 1 63A: m;>é_ n =' A mu + 1- IGII‘A.‘ OHM-z , I!) Mr”. «me MM ‘1' a: .. (“as , M1~ w 4:: - 4x =§(A+a+x[~33)4x Ho) ‘ “ ______ z A. \$9.32 Ho): (4+a)a+ 5‘ f, , . I . a w_ ‘ F60 ‘ .; X z A : ~—— -—-‘~ 9(4) . ’ (“53x + a; ram] _' M 9 F = 0 NM :‘1 a“‘ JV xal we haw F ze‘eg * " ‘ﬂT—M Wham—L131 Note that for any observables, we have: am) i dt 3 ([H. Al) + at a (A) ~— If A is intrinsically time independent mean: 9&9- = 0. Now, remember that if [q.oz]-ica.wohm H )I ACIAOJZ—T According to our ﬁrst equation, we have: Thus, we get: _mi‘ﬁ = d, (H. Al , “,,,_w_ww..~y-.m—mmm ...
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