7636S ADVANCED QUANTUM MECHANICS Solutions Spring. Considr a thr dimnsional kt spac. If a crtain st of orthonormal kts, say, and 3 ar usd as th bas kts, thn th oprators A and B ar rprsntd by a b A a and B ib a ib in which both a and b ar ral. a Obviously A xhibits a dgnrat spctrum. Dos B hav a dgnrat spctrum as wll? b Show that A and B commut. c Find a nw (orthonormal st of bas kts which ar simultanous ignkts of both A and B. Spcify th ignvalus of A and B for ach of th thr ignkts. Dos your spcication of ignvalus compltly charactriz ach ignkt? Solution: a From th matrix rprsntation of B w can s that th kt is an ignvctor of oprator B with ignvalu b, i.. B b ib ib b b b. As th matrix of oprator B is Hrmitian so its ignvalus must b ral. It only rmains to diagonaliz th minor M of matrix B. thrfor dt (M λi, λ + b λ ±b. W hav found that th ignvalus of B ar { b, b, b}, concluding that oprator B has a dgnrat spctrum. b Lt us calculat th products AB and BA indpndntly. AB BA a a a b ib ib b ib ib a a a ab iab iab ab iab iab,.
W ar now in conditions of writing th commutator: A, B] AB BA. Thrfor A and B must shar a simultanous st of ignvctors. c W alrady hav th rst of th ignvctor in that particular st, i.. kt. Lt us nd now th rmaining ignvctors of oprator B in th subspac M. Eignvctor associatd to ignvalu b. Lt us rnam it as. ( ( ( ib c bc ib c 3 bc 3 Thus { ibc3 bc ibc bc 3 c 3 ic. If w want our ignvctors normalizd, thn c c 3 /. Eignvctor associatd to ignvalu b. Lt us rnam it as 3. ( ( ( ib c bc ib c 3 bc 3 Thus { ibc3 bc ibc bc 3 c 3 ic. If w want our ignvctors normalizd, thn c c 3 /. W hav to chck that ths nw ignvctors ar shard with oprator A. a A a a a a i ia A 3 a a a i a ia a 3 Th primd notation ( is nough, but anothr and sur mor informatic naming convcntion is to charactriz th ignvctors with thir ignvalus rspct oprator A and B, rspctivly a, b a, b ( + i 3 a, b ( i 3
. Evaluat th uncrtainty rlation of x and p oprators for a particl connd in an innit potntial wll (btwn two unpntrabl walls. Som hlp: In this cas th potntial can b writtn:v (x, whn < x < a and othrwis V. From quantum mchanics w rmmbr that th wav function in such a potntial rads ψ n (x /a sin (nπx/a, in which numbr n rfrs to th nth xcitation whil n is th ground stat. Solution: Th uncrtainty rlation for x and p is givn by th product of th standard dviations x and p, i.., x p. Th standard dviation for a gnric obsrvabl q in th systm stat ψ is givn by ψ q q ψ q ψ. In our cas, w know that for a potntial wll in which V in th rang < x < a, th ignfunctions for this particular problm can b writtn as ( nπx ψ n (x a sin. a Knowing this, w ar rady to calculat th xpctation valus of obsrvabls x, x, p and p. a x ψn(xxψ n (x dx a x sin (nπx/a dx a a 4 a, a x ψn(xx ψ n (x dx p a 3 a 6 ( 3 π n a 3 ψn(x( i d dx ψ n(x dx a a x sin (nπx/a dx ( 3, π n i sin(nπx/anπ/a cos(nπx/a dx, a p ψn(x( d dx ψ n(x dx n π a sin (nπx/a dx a a n π a a a π n /a In th valuation on nds partial intgration and doubl angl formula: sin x /( cos x. Subtituting th abov valus into th dnition for th uncrtainty rlation, w obtain that x p π n 6 > 3 n 6 > 9 6 3 3 3.
3. Show that p x α i p p α and β x α dp φ β(p i p φ α(p. Solution: Hr w nd th rprsntation of ignstat x in th momntum spac p x xp (i p x, π also w nd th hrmicity of x ( x x α x x α and th familiar dirntation rul (xp(ax x a xp(ax. p x α p dx x x x α dx x p x x α }{{} I dx x xp ( i p x x α π dx i xp ( i p x x α p π i p dx x x p α i p p α Th scond rsult is a corollary of th rst on: β x α β dp p p x α }{{} I dp β p p x α dp φ β(p i p p α dp φ β(p i p φ α(p.
4. Considr spin prcssion of lctron in static uniform magntic ld in th z dirction and calculat th xpctation valus of spin at tim t in y and z dirctions whn th initial stat of th systm at t is S x ; S z ; + S z ;. Solution: W considr magntic ld to b B Bẑ, so that th Hamiltonian is writtn as ( B H µbs S z ω c S z. m c Th tim volution oprator for this systm is U(t, xp ( is z ω c t/, whr ω c B/m c. Th xpctation valu for S z is thn calculatd as follows S z (t S x ; U (t, S z U(t, S x, iωct/ S z ; + iωct/ S z ; ] S z iω ct/ S z ; + iωct/ S z ; ]. Calculating th xpctation valu of S y is quivalnt: S y (t S x ; U (t, S y U(t, S x, iωct/ S z ; + iωct/ S z ; ] S y iωct/ S z ; + iωct/ S z ; ] i 4 sin (ω ct. iω ct/ S z ; + iωct/ S z ; ] iω ct/ S z ; iωct/ S z ; ] It is trivial to s that spin prcds in th xy-plan with a frquncy ω c and with no projction into th z axis. Anothr and mor straightforward way to calculat th xpctation valus is to apply th matrix rprsntations of S i drivd in prvious xrciss and rprsnt also th tim dpndnt stat in th ignbasis of S z. S x ; ; t U(t, S x ; ; t ( iωct/ iωct/ S x (t S x ; ; t S z S x ; ; t ( ( ( iωct/ iωct/ iωct/ iωct/ S y (t S x ; ; t S z S x ; ; t ( ( ( iωct/ iωct/ i iωct/ i iωct/ ( ( iωct/ iωct/ i iωct/ i iωct/ sin(ω ct