Teoretisk Fysik KTH. Advanced QM (SI2380), test questions 1

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1 Teoretisk Fysik KTH Advanced QM (SI238), test questions NOTE THAT I TYPED THIS IN A HURRY AND TYPOS ARE POSSIBLE: PLEASE LET ME KNOW BY IF YOU FIND ANY (I will try to correct typos asap - if you find a typo several days after this has been published, please check the course homepage if the typo you found has not yet been corrected before sending me a ). Abbreviations: [S] = J J Sakurai: Modern Quantum Mechanics, Second Edition (Pearson, 2) [S] P.8 = Problem.8 in [S] (etc.); [S] Ch..2 = Chapter.2 in [S] (etc.); [S] Eq. (.3.39a) = Equation (.3.39a) in [S] (etc.) In the second midterm test you will get questions similar to the following (to check that you digested the material in [S] we discussed in the course). Some hints on how to answer these questions are given at the end of this text. All problems are on [S] Ch. 5.. Find the eigenvalues of the matrix C A + λb, A = 2, B = and λ a small coupling constants, in second order perturbation theory. Compute also the corresponding eigenvectors in first order perturbation theory. 2 Hints: 3 Let H = H + λv, with V n,m n V m and H n = E n n, (a) If E m E m m n : E n = E n + λv n,n + λ 2 m n, n n =, n, m = δ n,m. n V n,m V m,n +... En Em V m,n n = n + λ m E m n n Em (b) If En En = En 2 = = En g, Em En m n j : g g H eff = En n j n j + λ V nj,n k n j n k. j= Version (v) by Edwin Langmann on September 25, 2; revised by EL on September 27, 2 2 A typical midterm question would be to compute one of these eigenvalues to second order- of one of these eigenstates. 3 In any possible questions on perturbation theory you will be given the information below. j,k=

Abbreviations: [S] = J J Sakurai: Modern Quantum Mechanics, Second Edition (Pearson, 2) [S] P.8 = Problem.8 in [S] (etc.); [S] Ch..2 = Chapter.2 in [S] (etc.); [S] Eq. (.3.39a) = Equation (.3.39a) in [S] (etc.

2 2. Find the eigenvalues of the matrix C A + λb, A =, B = and λ a small coupling constants, in first order perturbation theory. Hints: Same as in Problem. 3. Use the variational method to find the best approximation to the smallest eigenvalue and corresponding eigenvector of a matrix A using a given variational eigenvector v: ( ) ( ) 3 A =, v = 3 7 α with α a real variational parameter. 4. Use the variational method to find the best approximation to the smallest eigenvalue and the corresponding eigenvector of a matrix A using a given variational eigenvector v: A = with α a real variational parameter. 5. Given the Hamiltonian d 2, v = H = 2 + V (x), V (x) = 2m dx2 α, { if x < a otherwise of a quantum particle in one dimension and in the position representation. Use the variational function { a 2 x 2 if x < a u(x) = otherwise to estimate the groundstate energy of this Hamiltonian. Compare your result with the exact result. 6. Given the Hamiltonian d 2 H = 2 2m dx λδ(x) 2 of a quantum particle in one dimension and in the position representation, with λ >. Use the variational function u(x) = e α x, with α a variational parameter, to estimate the groundstate energy of this Hamiltonian. Compare your result with the exact result. 2

Use the variational method to find the best approximation to the smallest eigenvalue and corresponding eigenvector of a matrix A using a given variational eigenvector v: ( ) ( ) 3 A =, v = 3 7 α with

3 7. Use the variational method to find the best approximation to the groundstate energy and the corresponding groundstate of a Hamiltonian H using the given variational state : H = 2 j= ( ) t j j + + j + j λ, = + a with a real variational parameter a; t > and λ > are model parameters. 8. Let H be a Hamiltonian with groundstate E (= smallest eigenvalue of H). Prove that, for any state, H E. () Describe how this result can be used to find approximations to the groundstate and groundstate energy of H. You can assume that H has orthonormal eigenstates j with corresponding eigenvalues E j, j =,, 2, Consider the Schrödinger equation for a time dependent Hamiltonian H = H + V (t), i t α; t = (H + V (t)) α; t, α; t ) = α, (2) and assume that H has an orthonormal basis of eigenstates j with corresponding eigenvalues E j, j =, 2, 3,.... Show that (2) has the solution α; t = j c j (t)e ie jt/ j. (3) with c j (t) determined by the following equation, c j (t) = c j (t ) + k t ds e i(e j E k )s/ V jk (s)c k (s) (4) i t with V j,k (t) j V (t) k. Use this to derive the solution c j (t) in second order perturbation theory.. Consider the Hamiltonian H = E + E 2 2 2, V (t) = γ 2 e iωt + γ 2 e iωt with orthonormal kets j, j =, 2, and real parameters E 2 > E, γ, and ω >. Find the solution of the time dependent Schrödinger equation i t α; t = (H + V (t)) α; t, α; =, using first order time dependent perturbation theory. Use this result to compute the probability α; t 2 in leading non-trivial order. Give a physical interpretation of α; t 2. 3

() Describe how this result can be used to find approximations to the groundstate and groundstate energy of H.

4 Hint: First order time dependent perturbation theory: α; t = j c j (t)e ie jt/ j with c j (t) c j () + i and V j,k (t) j V (t) k. k t dse ie js/ V jk (s)e ie ks/ c k () 4

5 Hints. The eigenvalues of A are all different, and we thus can use non-degenerate perturbation theory. As an example, I only compute the eigenvalue c 2 = of C: c 2 = a 2 + λb 2,2 + λ 2 j 2 and the corresponding eigenvector v 2 e 2 + λ j 2 with e j the usual basis vectors. B 2,j B ( j,2 7 + = 2 + λ6 + λ 2 2 ) a 2 a j V j2 e j = e 2 + λ 7 a 2 a j 2 3 e 3 + λ 4 2 e 3 = 4λ 7λ Answer: c = + 3λ 57 2 λ2 +..., c 2 = 2 + 6λ 33λ , c 3 = 3 + 8λ λ λ 5λ/2 v = 4λ 5λ/2 +, v 2 = 7λ +, v 3 = 7λ + Remark: A useful check of the computation is to verify that the eigenvectors v j are indeed orthogonal in leading order in λ. 2. The third eigenvalue of C can be computed by non-degenerate perturbation theory. The first two eigenvalues of A are degenerate, and thus these eigenvalues have to be computed by diagonalizing ( ) + 3λ 4λ C eff = 4λ + 9λ ( ) ( λ 4 9 Answer: c = + λ +..., c 2 = + λ +..., c 3 = 3 + 8λ Compute E(α) v A v v v = + 6α 7α2 + α 2. and find the minimum by solving E (α) =. The latter equation has two solutions α = 3 and α 2 = /3, and one can check that the first solution corresponds to the minimum. Answer: The best approximation to the lowest eigenvalue a min of A is E(3) = 8, and a min 8. Remark: In this case the variational method gives the exact answer, which is not the case in general, of course. ) 5

.. 2 V j2 e j = e 2 + λ 7 a 2 a j 2 3 e 3 + λ 4 2 e 3 = 4λ 7λ Answer: c = + 3λ 57 2 λ2 +..., c 2 = 2 + 6λ 33λ 2 +..., c 3 = 3 + 8λ + 23 2 λ2 +.

6 4. In this case E(α) v A v v v = + 4α + 4α2 + α 2. which has a minimum at α = /2 with E( /2) =. Answer: The best approximation to the lowest eigenvalue a min of A is E( /2) =, and a min. Remark: The exact value of the smallest eigenvalue is (this is found by numerical diagonalization of A). 5. See [S], Ch. 5.4, in particular Eq. (5.4.4). 6. Note that in this case the exact groundstate has the form of the variational function, so in this example the variational method gives the exact answer (you computed the exact solution in a previous exercise). 7. Compute and find the minimum of E(a). E(a) H λ + 2ta = + a 2 Answer: The best possible approximation to the groundstate energy E is E min = 2t 2 /( 4t 2 + λ 2 λ) E. It is obtained at a = a min = ( 4t 2 + λ 2 λ)/(2t). 8. See [S] Ch See [S] Ch. 5.7, or (alternatively) my notes N (available on my logbook on the course homepage).. See [S] p or my notes N. The exact answer to this problem is in [S] Eq. (5.5.2). Answer: c (t), c 2 (t) i t dsγe iωs+i(e 2 E )s/ = γ ω ( ei ωt ) with ω (E 2 E )/ ω. Thus, in leading non-trivial order, α; t 2 = c (t) 2 = c 2 (t) 2 4γ2 sin 2 ( ωt/2) 2 ω 2. This is the probability that the system is in the state at time t >, assuming it is in this state at time t =. 6

Note that in this case the exact groundstate has the form of the variational function, so in this example the variational method gives the exact answer (you computed the exact solution in a previous

1 Variational calculation of a 1D bound state

1 Variational calculation of a 1D bound state TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK FÖRDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM MECHAN- ICS Kvantmekanik fördjupningskurs SI38 för F4 Thursday December, 7, 8. 13. Write on each page: Name,