2m + V ( ˆX) (1) 2. Consider a particle in one dimensions whose Hamiltonian is given by


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1 Teoretisk Fysik KTH Advnced QM SI2380), Exercise Consider prticle in one dimensions whose Hmiltonin is given by Ĥ = ˆP 2 2m + V ˆX) 1) with [ ˆP, ˆX] = i. By clculting [ ˆX, [ ˆX, Ĥ]] prove tht ˆX 2 E E ) = 2 2m 2) where is n energy eigenstte with eigenvlue E. Assume tht ll eneergy eigensttes re bound sttes. 2. Consider prticle in one dimensions whose Hmiltonin is given by Ĥ = ˆP 2 2m + mω2 2 ˆX 2 + λ 2 ˆX4 3) with ˆP nd ˆX the usul momentum nd position opertors stisfying [ ˆP, ˆX] = i m > 0, ω > 0, nd λ > 0). ) Use the Ehrenfest theorem to derive the time evolution equtions for ˆXt) nd ˆP t). b) Compre your result in ) with the corresponding clssicl equtions, i.e. the Hmiltonin equtions for the clssicl Hmiltonin corresponding to the Hmiltonin in 3): H cl p, q) = p2 2m + mω2 2 q2 + λ 2 q 4. 4) Show tht, for λ = 0 hrmonic oscilltor), the clssicl equtions for qt), pt) re identicl with the ones for Xt), P t). Hint: The Hmilton equtions re q = H cl p, ṗ = H cl q 5) with q dq dt etc. 3. Show tht, for generlized hrmonic oscilltor Hmiltonins H cl q, p) = N p 2 j 2m j + m 2 N A jk q j q k 6) j,k=1 1 Version 1 v1) by Edwin Lngmnn on Sept. 16, 2010; revised by EL on October 11, I thnk Klle Bäcklund for pointing our typos nd suggesting improvements. 3 [Skuri] Section 2, Problem 5. 1
2 with A jk ) rel, symmetric, positive mtrix, the Hmilton equtions for q j t), p j t) re identicl with the time evolution equtions for ˆX j t), ˆP j t) obtined with the Ehrenfest theorem nd the corresponding quntum Hmiltonin. Hints: The Hmilton equtions re q j = H cl p j, ṗ j = H cl q j 7) for j = 1, 2,..., N. The quntum Hmiltonin corresponding to H cl is Ĥ = H cl X, P) with [ ˆP j, ˆX k ] = i δ jk. Introduction to subsequent problems The following problems re on bosons nd fermions systems nd to digest the formlism developed in the lectures. As in the lectures we use the following nottion: for opertors A, B. We consider nd j with the following reltions, [A, B] AB BA for bosons fermions 8) bosons fermions cretion nd nnihiltion opertors j [ j, k ] = δ j,k [ j, k ] = [ j, k ] = 0, 9) nd vcuum sttes Ω such tht j Ω = 0 j, Ω Ω = 1. 10) 4. Consider system of fermions on three sites, i.e. the one prticle sttes re j for j = 1, 2, 3, nd there re fermion opertors ) j nd vcuum Ω stisfying the reltions in 9) nd 10) for j, k = 1, 2, 3. ) Consider the oneprticle sttes for j = 1, 2, 3. Compute ψ 1) j ψ 1) k. b) Consider the twoprticle sttes ψ 1) j j Ω 11) ψ 2) jk j k Ω. 12) Compute ψ 2) jk ψ2) lm. c) Determine the dimension of the zero, one, two, nd three prticles sttes. Show tht the totl number of sttes in this model is 2 3. Hint: It is convenient to introduce symbolic nottion N 1, N 2, N 3 1) N 1 2) N 2 3) N 3 Ω 2
3 with N j = 0, 1. Convince yourself tht this is bsis of sttes. d) Find ll normlized eigensttes nd eigenvlues of the Hmiltonin E j j j 13) with rel constnts E j. e) Find ll normlized eigensttes nd eigenvlues of the Hmiltonin ) 14) with t > 0. Hint: Write the Hmiltonin s H 1 = j,k A jk j k nd digonlize the symmetric 3 3 mtrix A = A jk ). Introduce ã j = k f j ) k k, ã j = k f j ) k k with f j the normlized eigenvectors of A f j ) k is the kth component of f j ). Show tht these opertors obey the sme nticommuttor reltions s the opertors j, nd tht they llow to write H 1 = E j ã jãj. 5. Generlize the solution of the previous problem from 3 to n rbitrry number n sites: the one prticle sttes re j nd fermion opertors ) j now re for j = 1, 2,..., n with finite n. ) Consider the one nd two prticle sttes s in 23) nd 24). nd ψ2) ψ 1) j ψ 1) k jk ψ2) lm. b) Consider the Nprticle sttes Compute ψ N) j 1 j 2 j N Ω 15) with N = 0, 1,..., n. Prove tht the sclr product ψ N) k 1,k 2,,k M ψ N) 16) is nonzero only if N = M, nd if this is the cse it is equl to { ± if k 1, k 2,..., k N ) is n even odd permuttion of j 1, j 2,..., j N ). 17) 0 otherwise 3
4 c) Show tht the dimension of the Nfermion subspce is ) n N = n!. Show N!n N)! tht the dimension of the full fermion Fock spce is 2 n. d) Find ll normlized eigensttes nd eigenvlues of the Hmiltonin E j j j 18) with rel constnts E j. e) Find ll normlized eigensttes nd eigenvlues of the Hmiltonin j j+1 + j+1 ) j 19) with t > 0 nd ) n+1 ) 1 periodic boundry conditions). 6. The soclled tv model is defined by the Hmiltonin H = t j j+1 + j+1 ) j + V j j j+1 j+1 20) with fermion cretion nd nnihiltion opertors ) j nd ) n+1 ) 1 nd constnts t > 0 nd V > 0. ) Give physicl interprettion of this Hmiltonin. b) Find ll eigensttes nd eigenvlues of this Hmiltonin for n = 3. Hint: Show tht H commutes with the prticle number opertor ˆN = j j 21) nd conclude tht H hs eigensttes with fixed prticle number. 2prticle eigensttes e.g.) compute the mtrix elements To find the H 2) jk,lm ψ2) jk H ψ2) lm. 22) with jk, lm = 12), 13), 23) 3 3 mtrix). A convenient wy to find this mtrix is to compute H ψ 2) lm using the reltions in 9) nd 10). You cn find the eigensttes nd eigenvlues by digonlizing this mtrix. You will find tht the interction is irrelevnt for n = 3. c) Find ll eigensttes nd eigenvlues of this Hmiltonin for n = 4. Hint: The nontrivil prt of this problem is to find the twoprticle eigenfunctions. Show tht this corresponds to digonlizing 6 6 mtrix. 4
5 7. Consider system of bosons on three sites, i.e. the one prticle sttes re j for j = 1, 2, 3, nd there re boson opertors ) j nd vcuum Ω stisfying the reltions in 9) nd 10) for j, k = 1, 2, 3. ) Consider the oneprticle fermion sttes for j = 1, 2, 3. Compute ψ 1) j ψ 1) k. b) Consider the twoprticle boson sttes ψ 1) j j Ω 23) ψ 2) jk j k Ω. 24) Compute ψ 2) jk ψ2) lm. c) Determine the dimension of the Nboson sttes for N = 0, 1, 2, 3,.... Hint: It is convenient to introduce symbolic nottion N 1, N 2, N 3 1) N 1 2) N 2 3) N 3 Ω with N j = 0, 1, 2,.... Convince yourself tht this is bsis of Nboson sttes if you restrict to N j such tht j N j = N. d) Find ll normlized eigensttes nd eigenvlues of the Hmiltonin E j j j 25) with rel constnts E j. e) Find ll normlized eigensttes nd eigenvlues of the Hmiltonin ) 26) with t > 0. Hint: Use the sme method s in the corresponding fermion problem. 8. Generlize the solution of the previous problem from 3 to n rbitrry number n sites: the one prticle sttes re j nd boson opertors ) j now re for j = 1, 2,..., n with finite n. ) Consider the one nd two prticle sttes s in 23) nd 24). ψ 1) nd ψ2) j ψ 1) k jk ψ2) lm. b) Consider the Nprticle sttes Compute ψ N) j 1 j 2 j N Ω 27) with N = 0, 1,..., n. Prove tht the sclr product ψ N) k 1,k 2,,k M ψ N) 28) 5
6 is nonzero only if N = M, nd if this is the cse it is equl to { 1 if k 1, k 2,..., k N ) is permuttion of j 1, j 2,..., j N ) 0 otherwise. 29) d) Find ll normlized eigensttes nd eigenvlues of the Hmiltonin E j j j 30) with rel constnts E j. e) Find ll normlized eigensttes nd eigenvlues of the Hmiltonin j j+1 + j+1 ) j 31) with t > 0 nd ) n+1 ) 1 periodic boundry conditions). 6
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