Physics 428: Quantum Mechanics III Prof. Michael S. Vogeley Practice Problems 1

Transcription

1 Physics 428: Quantum Mchanics III Prof. Michal S. Vogly Practic Problms 1 Problm 1 A particl in fr spac in on dimnsion is initially in a wav packt dscribd by ( ) α 1/4 ψ(x) = αx 2 /2 a) What is th probability that its momntum is in th rang (p, p + dp)? (Hint: rmmbr th Fourir transform rlation btwn position and momntum spac). Th probability that th particl has momntum in this rang is φ(p) 2 dp, whr φ(p) is th Fourir transform of ψ(x), φ(p) = (2 h) 1/2 dx ψ(x) ipx/ h = (2 h) 1/2 dx ( α ) 1/4 αx 2 /2 ipx/ h To do th intgral, complt th squar in th xponnt, αx ipx h = α ( x + ip + 2 α h)2 α p 2 2 α 2 h 2 Thus ( ) α 1/4 φ(p) = (2 h) 1/2 p2 2α h dx α 2 (x+ip/α h)2 Th substitution u = α/2(x + ip/α h), du = dx α/2 yilds a simpl intgral ovr a Guassian, and w obtain 1 φ(p) = h p2/2α h2 α so th probability is φ(p) 2 dp = 1 h /α h2 p2 dp α b) What is th xpctation valu of th nrgy? Can you giv a rough argumnt, basd on th siz of th wav function and th uncrtainty principl, for why th answr should b roughly what it is? E = ψ H ψ = dx ψ Hψ (1) 1

2 = dx = α h2 1 2m = h2 α 4m ( ) α 1/4 αx 2 /2 [ h2 2 ] (α ) 1/4 αx 2 /2 2m x 2 (2) du (1 u 2 ) u2 (3) whr w mak th substitution u = αx in th third lin (look up ths intgrals if you don t undrstand th rsult!). Th uncrtainty principl is p x h/2. Th width of th wav packt is x 1/ α, p = and sinc p p 2 p 2, th nrgy must b of ordr E p2 2m ( p)2 2m 1 2m which is within a factor of 2 of th xact rsult. ) 2 ( h α = h2 α 2 8m (4) Problm 2 Considr a fr particl of mass m in on dimnsion with priodic boundary conditions: ψ(x + ) = ψ(x) a) Writ down th complt st of normalizd nrgy ignfunctions and ignvalus. Th solution to th Schrödingr quation for a fr particl dfind on a intrval of lngth is simply ψ k (x) = 1 ikx But th condition ψ(x+) = ψ(x) constrains th allowd valus of k, bcaus w rquir ikx = ik(x+) = ikx ik or ik = 1, which implis k = 2n with n =, ±1, ±2,... Thus th allowd solutions ar ψ n (x) = 1 iknx with k n = 2 n, n =, ±1, ±2... whr th classical ground stat n = is not allowd. Ths solutions ar clarly ignfunctions of both H and p, 2 Ĥψ n (x) = h2 2m x 2 [ 1 iknx ] = h2 k 2 n 2m ψ n(x) 2

3 Thus, th ignvalu for th nth ignstat is h 2 k 2 n 2m = 22 h 2 m 2 n2 Similarly, ˆpψ n (x) = i h [ ] 1 iknx = hk n ψ n (x) x Thus, ths ar also ignfunctions of momntum, with ignvalus hk n = 2 h n b) Show that any two of ths ignfunctions corrsponding to diffrnt ignvalus ar orthonormal; that is dx ψm(x)ψ n (x) = δ nm Plugging in th solution abov, dx ψ m(x)ψ n (x) = 1 If m = n, th intgral is trivially qual to 1. For m n, dx i(kn+k+m) = dx i(kn km) 1 [ i 2 (n m)x] i(k n k m ) = Thrfor, dx ψ m(x)ψ n (x) = δ nm Problm 3 A particl of mass m movs in th potntial V (x) = { 1 2 kx2 x > x What ar th nrgy lvls and ignfunctions for th this systm? (Hint: compar this problm to that of th simpl harmonic oscillator.) This potntial is a smi-infinit harmonic oscillator. For x >, th potntial is idntical to that of th S.H.O. Th infinit potntial at x = rquirs th ignfunctions to b zro at x =. This is what all th anti-symmtric (odd) ignfunctions of th S.H.O. 3

4 do. Thus, th ignfunctions of this problm ar proportional to th anti-symmtric ignfunctions of th S.H.O., with diffrnt normalization, 1 = dx ψ (x)ψ(x) instad of 1 = dx ψ (x)ψ(x) = 2 dx ψ (x)ψ(x) Th ignfunctions ar thrfor ψ n (x) = ( 2un (x) n = 1, 3, 5,... x > x whr th u n (x) ar th S.H.O. ignfunctions. In trms of th variabl y = x mω/ h, th ignfunctions ar 1 ψ n (y) = 2 n 1 n! y2 H n (y) whr th H n (y) ar Hrmit polynomials. Th nrgy lvls ar E n = (n + 1/2) hω, n = 1, 3, 5,... Problm 4 In thr-dimnsional spac, rotation of a vctor about th z axis is prformd by th matrix cos φ sin φ R(φ) = sin φ cos φ 1 What ar th ignvalus of this matrix? What is thir magnitud? Th ignvalu quation is cos φ sin φ sin φ cos φ 1 which is solvd by computing th dtrminant a b c = λ cos φ λ sin φ sin φ cos φ λ 1 λ a b c = and finding th roots λ 1 = 1, λ 2,3 = ±iφ. Th magnitud of all th ignvalus is λ = 1. 4

5 Problm 5 Considr th matrix Q = 1 i 1 i 1 a) Is Q Hrmitian? Ys. Complx conjugat all th matrix lmnts and rvrs th indics and you gt th sam matrix (Q = Q). b) What ar th ignvalus of Q? As in problm 4, th ignvalus ar found by computing th dtrminant 1 λ i 1 i λ = 1 λ and finding th roots of th rsulting polynomial. Th ignvalus ar λ =, 1, 2. Problm 6 Considr th angular momntum matrics in th basis of sphrical harmonic ignfunctions. That is, th matrix lmnts of 2 ar givn by and th matrix lmnts of z ar givn by Y l m 2 Y lm Y l m z Y lm Notic that th full matrix can b dcomposd into submatrics (corrsponding to angular momntum of dimnsion 2l + 1): a 1 1 submatrix for l =, a 3 3 submatrix for l = 1, tc. Writ out th matrics for 2 and z up to and including l = 2 in this rprsntation. Indicat th submatrics by dashd lins. Rcall 2 Y lm = l(l+1) h 2 Y lm and th orthonormality of th sphrical harmonics, which yilds Y l m 2 Y lm = l(l + 1) h 2 δ ll δ mm Similarly, z Y lm = m hy lm yilds Y l m z Y lm = m h 2 δ ll δ mm 5

6 Thrfor, th 2 matrix is 2 = 2 h 2 2 h 2 2 h 2 6 h 2 6 h 2 6 h 2 6 h 2 6 h 2... and th z matrix is z = h h 2 h h h 2 h... 6