Physics 5153 Classical Mechanics. The Virial Theorem and The Poisson Bracket-1

Transcription

1 Physics 5153 Classical Mechanics The Virial Theorem an The Poisson Bracket 1 Introuction In this lecture we will consier two applications of the Hamiltonian. The first, the Virial Theorem, applies to systems that are boune. It allows us to etermine statistical properties of systems. The other is the Poisson bracket, which has allows us to etermine properties of systems without solving the equations of motion, an is connecte to the commutator in quantum mechanics. 1.1 The Virial Theorem Let s consier the variable Q p i q i, which remains boune if both p i an q i are finite. From Hamilton s equations of motion we have for the total erivative of Q Q p i q i q i ṗ i p i q i (1) so that Q(t) Q(0) 1 T Qt 1 T ( ) t p i q i (2) T T 0 T 0 If the system is perioic an we integrate over one perio, then left han sie is zero. If the system is boune, then as T the left han sie goes to zero. In these case, we have p i q i (3) where the brackets enote the time average values of the enclose quantities. The quantity V q i (4) is a generalization of the virial function introuce into the kinetic theory of gases by Clausius 1. Thus the time average virial function is V p i (5) For a system of N particles, the right-han sie becomes p i q i 2 T i 2N T (6) where the bar enotes an average over time an kinetic energy of each particle. The virial function for this system can be also expresse as follows V ṗ i q i I P r ˆnS (7) 1 This is just the total work one on the gas, incluing external an internal forces. The Virial Theorem an The Poisson Bracket-1

2 where the left ha sie is the work one by the intermolecular forces, an the secon term is the work one in containing the gas with in the container. Using the ivergence theorem, we arrive at V I V P r I 3P V (8) where V is the total volume. Combining the two expression for the average virial function we arrive at P V 2 3 N T I 3 Nkθ I 3 where θ is the absolute temperature from the expression T 3 2kθ. If I 0, this is the ieal gas law. Another application of the virial theorem is the case of a particle in a central potential V (r). In this case the average virial function is given by V p r p r (10) which is reuce to For a potential of the form V (r) Kr n, this expression becomes 2 T r V (r) (11) T 1 n V (r) (12) 2 For the gravitational or Coulomb forces n 1. This means that the average kinetic energy is 1/2 the average potential energy. 1.2 The Poisson Bracket The Poisson bracket appears often in classical mechanics an translates itself to quantum mechanics as the commutator. Let s consier the total time erivative of a function u that epens on the canonical variables q an p, an can also epen on the time t u t ( u q i u ) ṗ i u Using the canonical equations of motion, the total time erivative becomes u t ( u u ) u which can be written using the following shorthan notation where we efine the Poisson bracket as [u, H] qi,p i u t [u, H] q i,p i u ( u u ) (9) (13) (14) (15) (16) The Virial Theorem an The Poisson Bracket-2

3 Let s consier the case of u not epening explicitly on time, an the Poisson bracket being zero. In this case the quantity u(q, p) is a constant of the motion. This is the same as we have in quantum mechanics. If the commutator of an operator with the Hamiltonian is zero an oes not epen explicitly on time, then the operator correspons to a variable that is a constant of the motion. As an example of how this works, let s consier a simple harmonic oscillator in two imensions. Using the Poisson bracket, we will show that the angular momentum is conserve (a constant of the motion). The Hamiltonian for this system is H 1 2 (p2 1 p 2 2) 1 2 (q2 1 q 2 2) (17) where for simplicity we take m k 1. The canonical equations of motion are ṗ i q i The angular momentum associate with this system is The Poisson bracket is q i p i with i 1, 2 (18) J q 1 p 2 q 2 p 1 (19) J t J J (p 2 p 1 p 1 p 2 ) (q 2 q 1 q 1 q 2 ) 0 (20) therefore the angular momentum is a constant of the motion. Base on the efinition of the Poisson bracket, the canonical equations of motion can be expresse in a compact form. Let s start by calculating q i using the Poisson bracket q i [q i, H] q i (21) Similarly the canonical equation of motion for ṗ i can be erive ṗ i [p i, H] ṗ i (22) At this point we efine the Poisson bracket of any two functions that epen on the canonical variables ( u v [u, v] v ) u (23) If the functions we use are the canonical variables themselves, we get the following funamental brackets [q i, q j ] 0 [q i, p j ] δ ij [p i, p j ] 0 (24) Notice how closely these resemble the commutators in quantum mechanics. Just replace δ ij with i hδ ij. Before going further, let s consier a few properties of the Poisson bracket. Assume that we have the ynamical variables u(q i, p i ), v(q i, p i ), w(q i, p i ). The Poisson bracket has the following properties The Virial Theorem an The Poisson Bracket-3

4 1. Linearity: 2. Antisymmetric: 3. The prouct rule: 4. The Jacobi ientity: [αu βw, v] α [u, v] β [w, v] (25) [u, w] [w, u] (26) [u, wv] [u, w] v w [u, v] (27) [w, [u, v]] [u, [v, w]] [v, [w, u]] 0 (28) 5. The chain rule: if the variables are expresse in terms of a new set P i an Q i that are functions of the ol variables p i an qi, then the Poisson bracket of the ynamical variables is [w, u] w η i [η i, η j ] u η j (29) where η i Q i an η in P i for i 1 n an we have a total of 2n variables. 6. The funamental brackets: 1.3 Poisson s Theorem [q i, q j ] [p i, p j ] 0 [q i, p j ] δ ij (30) Assume that we have a set of ynamical variables an we woul like to know whether their time evolution is generate by a Hamiltonian or not. Poisson s theorem states that for the time evolution of a set of variables to be generate by a Hamiltonian, the variables must satisfy ] [ ] [Ṙ, t [R, S] S R, Ṡ Assume that the time evolution is generate by H(p i, q i, t), then accoring to Eq. 15 we have The first term can be reuce using the Jacobi ientity (31) t [R, S] [[R, S], H] [R, S] (32) [[R, S], H] [[R, H], S] [R, [S, H]] (33) The secon term can be expane as follows [R, S] ( R S R ) S ( 2 R S 2 R S ) ( R 2 S R 2 R ) [ R, S ] [ R, S ] (34) The Virial Theorem an The Poisson Bracket-4

5 Next we combine the two sets [R, S] t [ [R, H] R, S ] [ R, [S, H] S ] ] [ ] [Ṙ, S R, Ṡ which is the statement of the theorem Notice that this theorem also implies that if R an S are constants of the motion, then so is [R, S]. Let s consier two examples: The first we take the one-imensional simple harmonic oscillator. The two ynamical variables are q an p. These have the following equations of motion where we take m k 1. Applying Poisson s theorem we get (35) ṗ q q p (36) [q, p] 0 t (37) [ṗ, q] [p, q] [q, q] [p, p] 0 (38) which implies that the time evolution of the ynamical variables is generate by a Hamiltonian as expecte. As a secon example, consier the same two variables but with the following equations of motion Again we apply Poisson s theorem q pq ṗ pq (39) [q, p] 0 t (40) [ q, p] [q, ṗ] [pq, p] [q, pq] [q, p] p q [p, p] [q, p] q [q, q] p p q (41) which oes not satisfy Poisson s theorem, therefore the time evolution of these variables is not generate by a Hamiltonian. To show that a transformation is canonical, we take a set of ynamical variables R an S that are functions of q i an p i an transform the variables to a new set Q i an P i. Then if the Poisson bracket for both sets of phase coorinates are equal the transformation is canonical We start by taking the time erivative of both sies then expan this out [R, S] qi,p i [R, S] Qi,P i (42) t [R, S] q i,p i t [R, S] Q i,p i (43) ] t [R, S] Qi,P i [Ṙ, S Imposing the Poisson bracket conition, we get ] t [R, S] Qi,P i [Ṙ, S q i,p i Q i,p i [ ] R, Ṡ (44) q i,p i [ ] R, Ṡ (45) Q i,p i which accoring to Poisson s theorem, the time evolution of the variables are generate by a Hamiltonian. The Virial Theorem an The Poisson Bracket-5