Nov 03, 016
ASC, room A 38, phone 089-180410, email hartmut.ruhl@lmu.de Patrick Böhl, ASC, room A05, phone 089-1804640, email patrick.boehl@physik.uni-muenchen.de.
Das folgende Material ist nicht prüfungsrelevant. Das folgende Material dient der physikalischen Motivation der Betrachtung des Konzepts der Dichteoperatoren. Das folgende Material dient der Wiederholung elementarer Konzepte aus der Quantenmechanik vieler Teilchen.
Im Quantenfall lassen sich Mehrteilchensysteme mit Hilfe des Dichteoperators beschreiben. Der Dichteoperator gehorcht der Von Neumann-Gleichung. Für praktische Anwendungen benutzt man das Konzept der reduzierten Dichteoperatoren, welches dem reduzierter Wahrscheinlichkeitsdichten im klassischen Fall entspricht. Die dynamischen Gleichungen reduzierter Dichteoperatoren bilden eine BBGKY-Hierarchie wie im klassischen Fall. Aus der BBGKY-Hierarchie lassen sich unter bestimmten Voraussetzungen Einteilchendichteoperatoren im Gleichgewicht ableiten. In der vorliegenden Vorlesung spielen nur Dichteoperatoren im Gleichgewicht eine Rolle.
Density matrix and equation Today we derive the for reduced density operators to represent N-particle quantum systems. Let us assume that we have a Hilbert space of a N-particle system. Let ψ be a state vector in our Hilbert space. Let us define a density operator ρ ψ ψ, Tr (ρ 1, A Tr (ρ A. (1 With the help of our ρ the probability of finding the system in state ψ can be calculated. density operator ρ has an equation of motion i t ρ i t ( ψ ψ Hρ ρh +, ( where use of the Schrödinger equation has been made. For simplicity let us assume that we have a one-particle density operator in space representation. We find i t x ρ y x Hρ y x ρh + y (3 ( i t ρ(x, y, t H(x H + (y ρ(x, y, t. (4 As an example we calculate the average energy of the system E d 3 x x ρh x d 3 x d 3 y x ρ y y H x (5 d 3 x d 3 y H(x x ρ y δ 3 (x y d 3 x [H(x ρ(x, y, t] xy.
Wigner for free particles Let us next introduce a Wigner function defined by ρ (r η, r + η, t where W is the Wigner function. We have used x r η, y r + η, r x + y, η y x x 1 r 1 η, y 1 r + 1 η, m dp π e ipη W (r, p, t, (6 ( x y (7 r η. (8 m equation for the one-particle density operator in space representation yields H(x x m, 0 i t y m dp π i + x ρ (9 m (i t m r η e ipη W (r, p, t dp π e ipη ( t + pm r W (r, p, t. In a weak sense we find ( t + pm r W (r, p, t 0. (10
density matrix for free particles Let us assume that the quantum particles have an initial distribution W (r, p, 0 1 r r e p p 0 e 0. (11 πr 0 p 0 We obtain with the help of the equation of motion for the Wigner W (r, p, t 1 (r vt r e p p 0 e 0, v p πr 0 p 0 m. (1 We obtain for the density operator ρ (r η, r + η, t dp π e ipη W (r, p, t (13 dp 1 π e ipη e πr 0 p 0 ( r p m t r 0 p p e 0 1 1 π 3 r 0 + p 0 t m 4r +η p 0 r 0 +4iηr p 0 m t 4 r e 0 + p 0 t m.
Wigner for particles in a potential A one-particle Hamilton operator in an external field in space representation is We find where H(x H(y ( V r ± η H(x x m m dp + V (x. (14 ( r η + V r η ( V r + η, (15 π e ipη W (r, p, t (16 ( dp π e ipη V r ± p W (r, p, t. i
Wigner for particles in a potential equation becomes 0 i t dp π i [ x m dp π e ipη In a weak sense we find + V (x ] + y m + V (y ρ (17 (i t m ( r η V r η ( + V r + η e ipη W (r, p, t ( t + p m r + i [ ( V r ( p V r + ] p W (r, p, t. i i ( t + p m r + i [ ( V r ( p V r + ] p W (r, p, t 0. (18 i i This equation can be called, following our nomenclature, the quantum Vlasov equation for an ensemble of particles in an external force field. operators showing up in Wigner space are called pseudo differential operators. It is quite complicated to solve the above equation for an arbitrary potential V. With respect to a probabilistic interpretation of W one can ask if W is always a positive definite entity or if not what this would imply.
Density matrix for a quadratic potential Let us assume that V (x 1 D x. (19 This implies We obtain ( t + pm r D r p W (r, p, t 0. (0 r + D r 0 r(t A cos ωt + B sin ωt (1 m ω D m, r 0 A, p 0 mbω r(t r 0 cos ωt + p 0 mω sin ωt, p(t p 0 cos ωt r 0 mω sin ωt. We obtain r 0 r cos ωt p sin ωt ( mω p 0 p cos ωt + mωr sin ωt.
Density matrix for a quadratic potential Assuming we obtain W (r, p, t W (r, p, 0 1 r r e p p 0 e 0 (3 πr 0 p 0 ( r cos ωt p mω sin ωt 1 r e 0 πr 0 p 0 (p cos ωt+mωr sin ωt p e 0. (4 Transforming back to Hilbert space is left as an excercise.
question arises if there is a quantum BBGKY hierarchy as in the classical case. ensemble average of an N-particle operator A is defined as where A N is given by sums over one- and two-particle operators A; ρ N>0 Tr 1..N (A N ρ N, (5 N A N a(i + 1 N b(i, j (6 i1 i,j1,i j and the operator ρ N is the N-particle density operator. One can show that A; ρ Tr 1 (a(1 f (1 (1 + 1 ( Tr 1 b(1, f ( (1 (7 holds with f (1 (1 N 1 N Tr...N (ρ(1...n, (8 f ( (1 N N(N 1 Tr 3...N (ρ(1...n. (9
To proof the assertions we obtain for the ensemble average A; ρ (α 1 α...α N ρa α 1 α...α N (30 N>0 α 1...α N (α 1 α...α N ρ ᾱ 1...ᾱ N (ᾱ 1...ᾱ N A α 1 α...α N N>0 α 1...α N ᾱ 1...ᾱ N Next we define an N-particle state composed of a product of single particle states of non-interacting particles denoted by α 1...α N. N-particle state is either symmetric or anit-symmetric. This implies (ᾱ 1...ᾱ N a(i α 1...α N (ᾱ 1...ᾱ N a(1 α 1...α N, i (31 (ᾱ 1...ᾱ N b(i, j α 1...α N (ᾱ 1...ᾱ N b(1, α 1...α N, i, j. (3 Making use of the relations Eqn. (31 and (3 we can write A; ρ (α 1 α...α N ρ ᾱ 1...ᾱ N (33 N>0 α 1...α N ᾱ 1...ᾱ N ( N(N 1 (ᾱ 1...ᾱ N Na(1 + b(1, α 1 α...α N (α 1 f (1 (1 ᾱ1 (ᾱ 1 a(1 α 1 α 1 ᾱ 1 + 1 (α 1 α f ( (1 ᾱ1 ᾱ (ᾱ 1 ᾱ b(1, α 1 α, α 1 α ᾱ 1 ᾱ
where (α 1 f (1 (1 ᾱ 1 N (α 1 α...α N ρ ᾱ 1 α...α N, N 1 α...α N (34 (α 1 α f ( (1 ᾱ 1 ᾱ N(N 1 (α 1 α α 3...α N ρ ᾱ 1 ᾱ α 3...α N. N α 3...α N (35 Hence, to calculate ensemble averages with at most binary operators only the reduced distribution functions f (1 (1 and f ( (1 are required. For the reduced s-particle density operators f (s (1...s the equations hold, where i t f (s [ (1...s A(1...s, f (s ] (1...s s i1 Tr s+1 ([b(i, s + 1, f (s+1 ] (1..s + 1 (36 f (s (1..s N! (N s! Tr s+1...n (ρ(1...n. (37 N s To prove Eqn. (36 the starting point is the Von Neumann equation ( 1 i t ρ [A, ρ]. (38
When leaving the sum over N in Eqn. (5 away for a moment we obtain (α 1...α N i t ρ α 1...α N [ (α 1...α N A ᾱ 1...ᾱ N (ᾱ 1...ᾱ N ρ α 1...α N ᾱ 1...ᾱ N ] (α 1...α N ρ ᾱ 1...ᾱ N (ᾱ 1...ᾱ N A α 1...α N, (39 where an identity operation has been introduced. Next we rewrite the Hamiltonian A as A 1 a(i δ ij + 1 b(i, j (40 i,j ij 1 1 s s s N N s N N b(i, j + + + b(i, j ij i1 j1 i1 js+1 is+1 j1 is+1 js+1 1 s s s N N N + + b(i, j, i1 j1 i1 js+1 is+1 js+1
where b(i, j b(j, i and b(i, i 0 have been assumed i, j. Now we perform partial traces in Eqn. (39 ( i t α1...α sα s+1...α N ρ α 1...α s α s+1...α N (41 α s+1...α N [ (α1...α sα s+1...α N A ᾱ 1...ᾱ N (ᾱ 1...ᾱ N ρ α 1...α s α s+1...α N α s+1...α N ᾱ 1...ᾱ N ( ] α 1...α sα s+1...α N ρ ᾱ1...ᾱ N (ᾱ 1...ᾱ N A α 1...α s α s+1...α N. term on the left hand side of Eqn. (41 can be rewritten as ( (N s! i t f (s (1...s N! ( α1...α sα s+1...α N ρ α 1...α s α s+1...α N, (4 α s+1...α N where f (s N! ( (1...s α1...α sα s+1...α N ρ (N s! α 1...α s α s+1...α N. (43 α s+1...α N
Now we insert the partial sums from Eqn. (40. For the first one we obtain α s+1...α N ᾱ 1...ᾱ N ( 1 s s α 1...α sα s+1...α N b(i, j ᾱ 1...ᾱ N (44 i1 j1 (ᾱ 1...ᾱ N ρ α 1...α s α s+1...α N ( α 1...α sα s+1...α N ρ ᾱ 1...ᾱ N (ᾱ 1...ᾱ N 1 s s b(i, j α 1...α s α s+1...α N i1 j1 (α 1...α 1 s s s b(i, j ᾱ 1...ᾱ s ᾱ 1...ᾱs i1 j1 (ᾱ1...ᾱ sα s+1...α N ρ α 1...α s α s+1...α N α s+1...α N ( α1...α sα s+1...α N ρ ᾱ 1...ᾱ sα s+1...α N α s+1...α N (ᾱ 1...ᾱ 1 s s s b(i, j α 1...α s i1 j1 (N s! [ A(1...s, f (s ] (1...s, N!
where A(1...s (α 1...α 1 s s s b(i, j α 1...α s. (45 i1 j1 second contribution from the partial sum in Eqn. (40 yields ( s N α 1...α sα s+1...α N b(i, j ᾱ 1...ᾱ N (46 α s+1...α N ᾱ 1...ᾱ N i1 js+1 (ᾱ 1...ᾱ N ρ α 1...α s α s+1...α N ( α 1...α sα s+1...α N ρ ᾱ1...ᾱ N s N (ᾱ 1...ᾱ N b(i, j α 1...α s α s+1...α N i1 js+1 s [( α1...α sα s+1 (N s b(i, s + 1 ᾱ1...ᾱ s+1 i1 α s+1 ᾱ 1...ᾱ s+1 (ᾱ1...ᾱ s+1 α s+...α N ρ α 1...α s α s+1...α N α s+...α N ( α1...α s+1 α s+...α N ρ ᾱ1...ᾱ s+1 α s+...α N α s+...α N (ᾱ1...ᾱ s+1 (N s b(i, s + 1 α ] 1...α s α s+1
(N s! N! s i1 Tr s+1 ([b(i, s + 1, f (s+1 ] (1...s + 1, where f (s+1 (1...s + 1 (47 N! ( α1...α s+1 α s+...α N ρ α 1 (N (s + 1!...α s+1 α s+...α N. α s+...α N Use has been made of the relations (α 1...α N a(i α 1...α N (α 1...α N a(j α 1...α N, i, j, (48 (α 1...α N b(i, j α 1...α N (α 1...α N b(k, l α 1...α N, i, j, k, l. (49
last contribution from the sum in Eqn. (40 gives ( 1 N N α 1...α sα s+1...α N b(i, j ᾱ 1...ᾱ N α s+1...α N ᾱ 1...ᾱ N is+1 js+1 (50 (ᾱ 1...ᾱ N ρ α 1...α s α s+1...α N ( α 1...α sα s+1...α N ρ ᾱ 1...ᾱ N (ᾱ 1...ᾱ N 1 N N B(i, j α 1...α s α s+1...α N is+1 js+1 [( αs+1 α s+ (N (s + 1 (N (s + b(s + 1, s + ᾱ s+1 ᾱ s+ α s+1 α s+ ᾱ s+1 ᾱ s+ ( α1...α sᾱ s+1 ᾱ s+ α s+3...α N ρ α 1...α s α s+1α s+ α s+3...α N α s+3...α N ( α1...α sα s+1 α s+ α s+3...α N ρ α 1...α sᾱs+1ᾱ s+ α s+3...α N α s+3...α N (ᾱs+1 ᾱ s+ (N (s + 1 (N (s + b(s + 1, s + αs+1 α s+ ] 0 since the indices α s+1, ᾱ s+1 and α s+, ᾱ s+ can be exchanged in the second term of the last part of Eqn. (50. Taking all terms together Eqn. (36 is obtained. quantum BBKY-hierachy has exactly the same structure as the
As in the classical case a hierachy for the reduced density operators is obtained.
David B. Boercker and James W. Dufty, Degenerate Quantum Gases in the Binary Collision Approximation, Annals of Physics 119 43 (1979.