Practice Problems Solution

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4 Prctice Problems Solution Problem Consier D Simple Hrmonic Oscilltor escribe by the Hmiltonin Ĥ ˆp m + mwˆx Recll the rte of chnge of the expecttion of quntum mechnicl opertor t A ī A, H] + h A t. Let A ˆx x t x ī x, H] + h x ī x, p t h m + ] mw x i x, p ] m h i m h x, p] p + p x, p] i m h Hence position opertor not conserve for the system. Observe tht p m t x.. Let A ˆp i h x t p ī p, H] + h p ī p, p t h m + ] mw x i mw p, x ] h i mw h Hence momentum opertor not conserve for the system. Note i hp + pi h p m p, x] x + x p, x] imw h i hx xi h mw x Observe tht if the expecttion is tken over n energy eigenket, then m hw p n ˆp n n i m hw + n i n + n h x n ˆx n n mw h + + n i mw n + n hus the expecttion of the momentum n position vlues re conserve t zero in n energy eigenstte. Clssiclly, the hrmonic oscilltor hs to be in one of the continuous energy sttes. In ny of the continuous energy sttes, the prticle hs equl chnces in moving to the left n moving to the right. hus x p. More rigorously, without loss of generlity, let hen x x sin wt v x t wx cos wt v cos wt xt x sinwtt x pt mv coswtt p x coswt] w mv sinwt] w x cosπ cos] π mv sinπ sin] π Quntum mechniclly, the prticle nee not be in one of the energy eigensttes. It cn be in liner combintion of the energy eigensttes sy ψ c n + c n, where c, c C. Hence x n p is not conserve in generl. 3. Let AH Hence energy is conserve for the system. t H ī H, H] + h H t

5 . Differentite the result in prt n using the result in prt, t x m t p w x Hence x follows the clssicl eqution of motion of simple hrmonic oscilltor. 5. Recll the momentum opertor in terms of the ler opertors m hw ˆp i +, + n n + n +, n n n he expecttion vlue of the kinetic energy in given energy eigenstte is thus K.E n ˆp m hw m n n + n hw m n + n hw n n hw n n hw n n + n + n n + hw n + hw E n where n is the energy eigenket of the hrmonic oscilltor. hus the verge kinetic energy is hlf the constnt totl energy E n n + hw. 6. Similrly, recll the position opertor in terms of the ler opertors h ˆx mw + + he expecttion vlue of the potentil energy in given energy eigenstte is thus P.E n mwˆx n mw h mw n + + n hw n + + n hw n n hw n n hw n n + n + n n + hw n + hw E n Hence the verge potentil energy is lso hlf the constnt totl energy. In summry, we hve E n K.E + P.E with P.E K.E E n his is specil cse of the Viril heorem. his theorem provies generl eqution tht reltes the verge over time of the totl kinetic energy of stble system consisting of N prticles, boun by potentil forces, with tht of the verge totl potentil energy V. Mthemticlly, the theorem sttes N F k r k where F k represents the force on the kth prticle, which is locte t position r k. If the force between ny two prticles of the system results from potentil energy V r r n, i.e. proportionl to some power n of the inter-prticle istnce r, the viril theorem tkes the simple form n V Mterils extrcte from Wikipei Viril heorem. For simple hrmonic oscilltor V mw x. his mens n V E n s + V E n. 5

6 7. Alterntive wy to show result using result Let A xp in result. We hve t xp ī xp xp, H] + ī p h t h m + ] mw x, xp ī ] ] p h m, xp + mw x, xp ī p, xp ] + h m mw x, xp ] i x p, p ] + p, x ] p + mw x x, p ] + x, x ] p h m i p, x ] p + mw x x, p ] i h m h m p p, x] + p, x] p p + mw x x x, p] + x, p] x i i h p h m p + i hmw x p m mw x mw x m Now xp is time inepenent if the expecttion is tken over its energy eigenkets s xp Ψ xp Ψ ψnxe i En h t En i xpψ n xe h t x i h ψnxx x ψ nx] x t xp p As result mw x K.E. P.E.. Since K.E. + P.E. E n we hve m K.E. P.E. E n 8. Clim: Clssicl hrmonic oscilltor stisfy Viril heorem s well. Suppose prticle oscillte in clssicl hrmonic motion with mplitue x n ngulr frequency w. Without loss of generlity, let x x sin wt v x t wx cos wt v cos wt where v wx is the mximum spee. he verge kinetic energy over time is K.E mv t mv cos wtt mv ] π coswt + t mv sinwt w + t w mwv 8π π w mv mv E where π w is the perio of the ocsilltion n E mv mw x is the constnt totl energy. In similr mnner, the verge potentil energy over time is P.E mw x t mw x sin wtt mw3 x π mw3 x 8π cos wt t mw3 x t 8π π w mw x mw x sin wt w E ] π w Hence P.E K.E E n E K.E + P.E n clssicl hrmonic oscilltor stisfy Viril heorem s well. Problem Consier quntum mechnicl prticle of mss m in the groun stte of D infinite squre well of with. Its wve function is given by 6

7 n its energy eigenvlues is given by ψx sin π x π h E n n m for x for n,, 3,.... If the energy of the prticle is mesure, we will get E π h with probbiity. m. If the position of the prticle is mesure, the possible outcomes re x n the corresponing probbility ensity is ψx π sin x. 3. Possible outcomes of the momentum p hk R n P p hk for ll p R. o obtin the energy eigenfunction in momentum bsis, we tke the fourier trnsform of ψx, Ψ k k π h ψxe ikx x π h π sin x e ikx x π h e ik x e ikx i e ik kx ik k + e ik+kx ik + k i e ik kx e ik+kx x π h i π h k + ke ik kx + k ke ik+kx π h k k π h k k π h k k k + ke ik k + k ke ik+k k + k k k ] ] e ikx x k + k + k k e ik k ] k e ik + π h k k ] where k π eik e ik. Now π p h k k π h p h As result ] ] π Ψ k k h π h π h p e i p π h 3 h + π h p e ip h + Hence the infinite squre well groun stte energy eigenket in momentum representtion is Ψ p p π h 3 π h p e ip/ h + for p R 3 n the momentum probbility ensity function is Ψ p p Ψ pψ p π h 3 π h p ] e ip/ h + e ip/ h π h 3 + π h p ] e ip/ h + e ip/ h + π h 3 p ] π h p ] cos + h where the momentum p R. Note 7

8 i Although sin k x eikx e ikx n e ±ikx re eigenfunctions of the opertor ˆp i h i x, p hk is not the momentum of the prticle in n infinite squre well. he totl energy of the prticle is given by E n hw n h k n m. he energy sttes of the prticle re escribe by the wve number k n n not by momentum p. he reltion E p for the energy n momentum of the prticle in the box oes m not hol. his is becuse, the prticle is not free. here is potentil V in the system which is infinite outsie the bounry. For one thing, the energy spectrum is iscrete boun sttes while the momentum spectrum is continuous. he hmiltonin opertor however is still Ĥ ˆp m. ii Momentum opertor oes not exist in infinite squre well. he momentum opertor is efine s the genertor of trnsltion of the wvefunction on which it cts. Any shift of the wvefunction in n infinite squre well will violte this conition n hence it cnnot exist. On the other hn, we cn pply momentum opertor on free prticle moving long n infinite rel line or on the ngulr momentum of prticle moving in circulr motion. No violtion will result. iii he wvefunction Ψ p p in eqution 3 is the energy eigenfunction in momentum bsis. It cn help us to clculte the probbility of the groun stte prticle s momentum within some intervl. It is not n eigenfunction to some momentum opertor. Problem 3 Given P S x h the electron is in the eigenstte of S x h, i.e. ψ Expecttion vlue of S z in ψ is + ] ] + S z ψ ψ S z ψ + ] ] S z + h + ] ] h ] Expecttion vlue of S z in ψ is S z ψ S ψ z ψ + ] ] Sz + h + ] S z h + ] + ] h 8 + ] + ] h 8 ] + ] h Problem A prticle in n infinite squre well of with locte t x is in superposition of energy eigensttes escribe by the following wvefunction t t ψx, πx sin sin πx πx ] sin πx sin 7 7 sin ] πx + 7 sin 7 ψ x + 3 ] 3πx 7 ψ x + 7 ψ 3x Observe tht by. Hence ψx, is not normlise. Recll tht its energy eigenvlues is given π h E n n m n E for n,, 3,... Possible energy outcomes re E, E n E 3 with probbilities P E E /7 /7, P E E 8/7 /7 9, P E E 3 /7 /7 8

9 he verge energy of the prticle is 9 9 E E + E + E 3 E + E + 9E 6 E 3 b Normlise ψx, we hve π h m Ψx, ] 7 7 ψx, 7 ψ x ψ x + 7 ψ 3x ψ x+ 3 ψ x+ ψ 3 x P x / / Ψx, x / ψ + 9 ψ + ψ 3 + where in the bove clcultion, we hve use the fct tht 3 ψ ψ + 3 ψ ψ 3 + ψ ψ 3 x 3 + 3π + 3 5π π i / ψ nx s ψ n is symmetricl bout x / for ll n. ii For m n, / ψ m ψ n x / mπx sin m nπ sin m n πx sin ] nπx x m + nπ sin / cos m n πx m + n πx ] ] / π ] cos m + n πx { ] sin m n π m n ] x sin m + n π m + n ]} As result / ψ ψ x 3π, / ψ ψ 3 x 5π, / ψ ψ 3 x c ime evolution of the wve function Ψx, t ψ xe i E h t + 3 ψ xe i E h t + ψ 3 xe i E 3 h t Given nother wvefunction φx, t sin Φx, t sin 3πx 3πx rnsition probbility from stte Ψx, t to the stte Φx, t is e i E 3 h t. Normlizing we hve e i E 3 h t ψ 3 xe i E 3 h t P Ψ Φ Ψ Φ Ψ Φx ψ xe i E t h + 3 ψ xe i E t h + ] ] ψ 3 xe i E 3 h t ψ 3 xe i E 3 h t x ψ3x 9

10 Problem 5 A Hyrogen tom is in the following superposition of energy eigensttes φ nlm r A A 6 ssuming A R. b i Energy E E 3 with probbility. Ψr φ 3 r + 3 φ 3 r + Aφ 3 r ii Possible ngulr momentum L re ll + h, h n 6 h. iii Possible z-component of the ngulr momentum, L z re m h, h n h. c i E E 3 ii L + h h h iii L z + h 3 + h 6 3 h Problem 6 Puli mtrices he spin ngulr momentum opertors for S / spin re given by the following mtrices S x h, S y h i, S i z h i S x, S y ] h i i i i ] ii he hmiltonin of S / prticle in n externl mgnetic fiel is given by Given B Bẑ, we hve H gµ B H gµ B S B S x S y S z B Recll the eigenvlues of S z re h, h with eigenvectors eigenvlues of the prticle in the mgnetic fiel re gµ BB h n respectively. gµ B BS z n, gµ BB h h i i hs z respectively. Hence the energy with normlise eigenvectors iii Given Now S + S S + S x + is y h S S x is y h h h i + i i i i i, S +, S ] ] h h h h h h Hence S + n S ct s rising n lowering opertors respectively for the eigensttes.

11 Problem 7 A simple hrmonic oscilltor with chrge q is plce in perioic electric fiel. he hmiltonin escribing the system is given by H ˆp x m + kˆx + qe ˆx coswt Let A H in eqution in Problem, t H ī H, H] + h Hence the energy is not conserve not this system. H ˆp x t t m + ] kˆx + qe ˆx coswt t qe ˆx coswt] qe wˆx sinwt