Bogoliubov Transformation in Classical Mechanics

Transcription

1 Bogoliubov Tranformation in Claical Mechanic Canonical Tranformation Suppoe we have a et of complex canonical variable, {a j }, and would like to conider another et of variable, {b }, b b ({a j }). How do the equation of motion look like, if expreed in term of new variable? The anwer i given by the general relation iḃ = i{h, b } = j [ b b ]. (1) The next quetion i whether it poible to elect new variable in uch a way independently of the particular form of H that they would be alo canonical, that i (1) would be actually equivalent to iḃ =, (2) for any H. The anwer i poitive. To arrive at the neceary and ufficient condition for the new variable to be canonical we note that baically we require that [ b a j j a b ] j a j b [ a j b j a + a ] j j b. (3) Since the differential operator {/, / } are linear independent, Eq. (3) immediately lead to b = a j, b =. (4) Thee are the neceary and ufficient condition for {b } to be canonical. Problem 39. Show that the following two elementary tranformation dealing with only one variable, a j, are canonical. (a) Shift of the variable: a j a j + c j, where c j i a complex contant. (b) Shift of the variable phae: a j e iϕ j a j, where ϕ j i a real contant. (c) Show that the linear tranformation b = j u j a j i canonical, if and only if the matrix u i unitary. 1

2 A remarkable fact readily following from (4) i that the Poion bracket are invariant with repect to the canonical tranformation. That i the reult for {A, B} i the ame for any et of canonical variable. Problem 40. Prove thi. Actually, the invariance of the Poion bracket under ome tranformation of variable i a ufficient condition for that tranformation to be canonical. Indeed, noticing that for any quantity A we have A = i{a, a j }, A = i{a j, A}, (5) and requiring that the Poion bracket be invariant under our tranformation of variable, we immediately arrive at (4): b = i{a j, b } = a j, (6) b = i{b, a j } = i{a j, b } =. (7) Bilinear Hamiltonian. Bogoliubov Tranformation Given a Hamiltonian function, one can bilinearize it in the vicinity of it (local) minimum to tudy the normal mode. The general form of the Hamiltonian after bilinearization i (each variable i now reckoned from it equilibrium value a hift of a variable i a canonical tranformation): H = ( A ij a i a j B ij a i a j + 1 ) 2 B ij a i a j, (8) ij where without lo of generality we can aume that (the requirement for the Hamiltonian to be real and ymmetrization) A ij = A j i, B ij = B j i. (9) If a Hamiltonian of the form (8) feature a (local) minimum at the point a 1 = a 2 = a 3 =... = 0, then it can be diagonalized by linear canonical tranformation (Bogoliubov tranformation), b = j (u j a j + v j a j). (10) 2

3 By diagonalized we mean that in term of the new canonical variable {b } the Hamiltonian will read H = E b b, (11) that i will be equivalent to a et of non-interacting harmonic ocillator (normal mode). Before we how how to relate the u and v coefficient to the matrice A and B, let u firt etablih their general propertie. From the requirement that the tranformation (10) be a canonical we get [differentiate both ide with repect to a j, a j, b r, and b r and ue (4)]: [u j u rj v j vrj] = δ r, (12) j [u j v rj v j u rj ] = 0. (13) j Now we would like to make ure that the condition (12)-(13) are not only neceary, but alo ufficient for the tranformation to be canonical. To thi end we prove a ueful Lemma. If coefficient u and v atify (12)-(13), then the invere tranformation i given by a j = (ũ j b + ṽ j b ), (14) where ũ j = u j, ṽ j = v j. (15) Problem 41. Prove thi lemma by ubtituting a j from (14) into the right hand ide of (10). Now when we know how to expre a in term of b, we jut need to make ure that Eq. (4) are atified. And thi i eaily een from (15). The relation (12)-(13) for the tranformation (14) read [ũ j ũ k ṽ j ṽk] = δ jk, (16) [ũ j ṽ k ṽ j ũ k ] = 0. (17) 3

4 With (15) thee can be rewritten a [u j u k v j vk] = δ jk, (18) [u j v k v j u k] = 0. (19) Now we obtain the equation for the u and v coefficient. Eq. (11) implie E b =, (20) which become non-trivial if H i expreed in term of a. Differentiating both ide with repect to a j and a j, and taking into account b u j, b v j, (21) we get E u j = E v j = = =, (22). (23) Note that here it i legitimate to change the order of operator (/b) and (/a), becaue our tranformation i linear and, ay, (/b) are jut linear combination of (/a) : = j [ aj Calculating the derivative + a j ] = j [ ṽ j + ũ j ]. (24) = i = i [A ij a i + B ij a i ], (25) [A j i a i + B ij a i ], (26) and oberving that a i ṽ i = v i, a i ũ i = u i, (27) 4

5 we finally arrive at the ytem of equation: [A ij u i B ji v i ] = E u j, (28) i Uing the vector notation i [B ji u i A ji v i ] = E v j. (29) u = (u,1, u,2, u,3,...), v = (v,1, v,2, v,3,...), (30) and taking into account (9), we write thi ytem a A u B v = E u, (31) B u A v = E v. (32) We can alo write thi ytem in term of ũ and ṽ. Introducing the vector we have ũ = (ũ 1,, ũ 2,, ũ 3,,...), ṽ = (ṽ 1,, ṽ 2,, ṽ 3,,...), (33) A ũ + B ṽ = E ũ, (34) B ũ + A ṽ = E ṽ. (35) The energie of the normal mode, E, arie a the eigenvalue of the problem. The relation (12)-(13), which in the vector notation read are automatically guaranteed at E r E. Problem 42. Make ure that thi i the cae. u r u v r v = δ r, (36) v r u u r v = 0, (37) Eq. (31)-(32) are invariant with repect to the tranformation u v, v u, E E. (38) Thi mean that each olution ha it counterpart of the oppoite energy. However, only one olution of the pair i phyically relevant. Namely, the one with u u v v > 0. (39) 5

6 For it counterpart we then have u u v v < 0, (40) which mean that it i impoible to normalize it to unity, in contradiction with (36). Problem 43. Conider the one-mode Hamiltonian H = a a [ γ a a + γ a a ]. (41) (a) Show that without a lo of generality (up to a imple canonical tranformation) one may chooe γ to be real and poitive. (b) Explore the poibility of olving for the dynamic of thi model by Bogoliubov tranformation. (c) In the region of γ where Bogoliubov tranformation i not helpful, directly olve the equation of motion. Evolution a a Canonical Tranformation Remarkably, the evolution of a dynamical ytem can be conidered a a canonical tranformation. For the given et of variable {a j } we introduce another et {b j }, where, by definition, the value of b j i equal to the value of a j after ome fixed period of evolution, t. Firt we note that the definition i conitent, becaue given the particular form of the Hamiltonian, fixed time period t, and particular initial tate {a j }, we unambiguouly fix {a j (t)} by the equation of motion. Hence, each b j = a j (t) can be conidered a jut a function of all variable {a j }, the particular form of the function being related to the form of the Hamiltonian and the time period t. Now we make ure that our tranformation i canonical. It i enough to do it in the limit t 0, becaue then the cae of finite t can be viewed a jut a chain of infiniteimal canonical tranformation. Auming that t i arbitrarily mall, from the equation of motion we have and, correpondingly, b j = a j it + O(t 2 ), (42) b j a k = δ jk it 2 H a k + O(t 2 ), (43) 6

7 b j a k 2 H = it a + O(t 2 ). (44) k a j To check that the relation (4) do really take place (up to the term t 2 ), we note that = + O(t). (45) b j Problem 44. Finih the proof. If the Hamiltonian i bilinear, then the evolution equation i linear. Correpondingly, {a j (t)} i related to {a j (0)} by a linear tranformation, which can be written in the vector notation, a (a 1, a 2, a 3,...), a a(t) = U(t) a(0) + V (t) a (0). (46) Here U and V are ome time-dependent matrice the form of which i defined by the Hamiltonian only not by the initial tate a(0). The above-proven theorem tate that the tranformation (46) i canonical. But the linear canonical tranformation i nothing ele than the Bogoliubov tranformation. Hence, the evolution of a ytem with bilinear Hamiltonian i given by time-dependent Bogoliubov tranformation. Problem 45. Suppoe ome bilinear Hamiltonian i diagonalizable by the Bogoliubov tranformation (10). Expre the matrix element U jk (t) and V jk (t) in the equation (46) in term of Bogoliubov matrice, u j and v j, and eigenvalue, E. Problem 46. In the cae of a ingle-mode ytem with a bilinear Hamiltonian, Eq. (46) read a(t) = U(t) a(0) + V (t) a (0), (47) where U(t) and V (t) are ome function of time. Find U(t) and V (t) for the Hamiltonian (41) of the Problem 43. Stability of Equilibrium Solution By equilibrium olution of the equation of motion we mean the olution which doe not evolve in time:, ȧ 0. Thi immediately implie that the equilibrium olution correpond to uch a point, {a (0) }, in the pace of variable {a }, where : a = a = 0 (at a = a (0) ). (48) 7

8 We will alo refer to the equilibrium olution a equilibrium point. Eq. (48) mean that the equilibrium point are the point of extremal behavior of the Hamiltonian. In particular, the previouly dicued point of local minima are the equilibrium point. At the equilibrium point {a (0) }, a natural quetion arie of what happen if the tate of the ytem i lightly perturbed. The general way of anwering thi quetion i to expand the Hamiltonian in the vicinity of the equilibrium in term of the hifted variable a a a (0) (auming that their abolute value are mall enough). The reulting Hamiltonian i of the bilinear form (8), with A ij = 2 H a i, B ij = 2 H a i (at a = a (0) ). (49) Linear term of the expanion are zero becaue of Eq. (48) and the contant term i of no interet. The equation of motion for the Hamiltonian (8) i i t a = A a + B a. (50) Since thi i a linear equation, it general olution can be written a a um of elementary olution (the neceity of the term with complex conjugated ω come from the term with complex conjugated a) a = e iωt f + e iω t g. (51) Subtituting thi into (50) and eparating the term with e iωt and e iω t, which are linear independent at Re ω 0, we get If Re ω = 0, we write and obtain A f + B g = ω f, (52) B f + A g = ω g. (53) a = e λt f, (54) A f + B f = iλ f. (55) The form of the ytem of equation (52)-(53) i the ame a that of (34)-(35). And thi i not a coincidence. In the cae when ω are real the elementary olution (51) correpond to the normal mode found by Bogoliubov 8

9 tranformation. If at leat one of the frequencie ω i not real, then the equilibrium tate i dynamically untable with repect to mall perturbation. There will be a mode (with Im ω > 0) that will be increaing exponentially. But how doe one derive the exitence of Im ω > 0 from the fact of exitence of a complex ω? Why i it impoible to have Im ω < 0 for all complex ω? Actually, thi follow from the fact that the evolution of the ytem can be viewed a the Bogoliubov tranformation. Suppoe we have ome complex ω = ω 0 + iλ: { } a(t) = e iω0t f + e iω0t g e λt. (56) On the other hand, inverting the Bogoliubov tranformation (46), we have where, in accordance with (15), Combining (56) and (57), we get a(0) = a(0) = Ũ a(t) + Ṽ a (t), (57) Ũ = U, Ṽ = V T. (58) { } e iω0t Ũ f + e iω0t Ũ g + e iω0t Ṽ f + e iω0t Ṽ g e λt. (59) The left-hand ide of Eq. (59) i t-independent, while the right-hand ide contain the global exponential factor e λt. Hence, either Ũ, or Ṽ, or both hould contain the compenating factor e λt. But then, by Eq. (58), the ame factor hould be preent in U, or V, or both. Hence, there hould exit an elementary olution containing thi factor. Apart from the dynamic intability, when the olution grow exponentially, there can alo be a tatitical intability. In the cae of tatitical intability, all ω are real, but not all of them are poitive. The Hamiltonian thu can be diagonalized by the Bogoliubov tranformation, and the dynamic within the bilinear Hamiltonian i jut the uperpoition of ocillating normal mode. However, in a typical real ituation, when there are higher order term in the Hamiltonian (ay, an interaction with a heat bath) the amplitude of all the normal mode will be gradually drifting to higher and higher value, ince thi lead to the increaing entropy without violating energy conervation: The poitive contribution to the total energy from the poitive-e mode i compenated by negative contribution from the negative-e mode. 9