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Statistical Description of Macroscopic Systems of Particles

Statistical Description of Macroscopic Systems of Particles. Now, we are ready to talk about PHYSICS In the rest of the course, we’ll combine statistical ideas with the Laws of Classical or Quantum Mechanics ≡ Statistical Mechanics

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Statistical Description of Macroscopic Systems of Particles

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  1. Statistical Description of MacroscopicSystems of Particles

  2. Now, we are ready to talk about PHYSICS • In the rest of the course, we’ll combine statistical ideas with the Laws of ClassicalorQuantum Mechanics≡ Statistical Mechanics • We can use either a classical or a quantum description of a system. Of course, which is valid obviously depends on the problem!!

  3. Four Necessary Ingredientsfor a Statistical Description of a Systemwith many particles: 1. Specify the System“Macrostate”. 2. Choose a Statistical Ensemble 3. Formulate a Basic Postulate aboutà-priori Probabilities. 4. Do Probability Calculations

  4. 1. Specify the System“Macrostate”. • From your undergrad course, you should recall what is meant by “Macrostate” & that this is very different than the system “Microstate”! We’ll quickly review these concepts. • Macrostate Macroscopic System State A specification of the system’s macroscopic parameters

  5. Specification of the System State Microstate Microscopic System State • A quantum description of the System:This means specifying a (large!) set of quantum numbers. • Classical Description of the System: This means specifying a point in a large dimensional phase space.

  6. Quantum Description of the System: • For an isolated system, this means specifying a subset of the quantum states of the system. • The system is described by macroscopic parameters (that can be measured).

  7. 2. Statistical Ensemble: • We need to decide exactly which ensemble to use. This is also discussed in this chapter. In either Classical Mechanics or Quantum Mechanics: • If we had a detailed knowledge of all positions & momenta of all system particles & if we knew all inter-particle forces, we could (in principle) set up & solve the coupled, non-linear differential equations of motion, we could find EXACTLY the behavior of all particles for all time!

  8. If we could set up & solve the coupled, non-linear differential equations of motion, we could (in principle) find EXACTLY the behavior of all particles for all time! • In practice we don’t have enough information to do this. Even if we did, such a problem is Impractical, if not Impossible to solve! • Instead, we’ll use Statistical/Probabilistic Methods.

  9. Statistical/Probabilistic Methods: Require choosing an Ensemble • Now, lets think of doing MANY (≡ N) similar experiments on the system of particles we are considering. In general, the outcome of each experiment will be different. • So, we ask for the PROBABILITY of a particular outcome. This PROBABILITY≡ the fraction of cases out of N experiments which have that outcome. • This is how probability is determined by experiment. A goal of STATISTICAL MECHANICS is to predict this probability theoretically.

  10. Next, we need to start somewhere, so we need to assume 3. A Basic Postulateabout à-prioriProbabilities. • “à-priori” ≡ prior(based on our prior knowledge of the system). • Our knowledge of a given physical system leads is to expect that there is NOTHING in the laws of mechanics (classical or quantum) which would result in the system preferring to be in any particular one of it’s Accessible (micro) States.

  11. Webster’s on-line Dictionary: Definition of“à-priori” 1a:Deductive 1b:Relating to or derived by reasoning from self-evident propositions. a synonymto“à-posteriori” 1c: Presupposed by experience. 2a: Being without examination or analysis. analysis :Presumptive 2b: Formed or conceived beforehand

  12. 3. Basic Postulate about à-priori Probabilities. • There is NOTHING in the laws of mechanics (classical or quantum) which would result in the system preferring to be in any particular one of it’s Accessible Microstates.

  13. 3. Basic Postulate about à-priori Probabilities. • So, (if we have no contrary experimental evidence) we make the hypothesisthat: it is equally probable (or equally likely) that the system is inANY ONE of it’s accessible microstates.

  14. The hypothesisis that it is equally probable (equally likely) that the system is in ANY ONEof it’s accessible microstates. • This postulate is reasonable & doesn’t contradict any laws of mechanics (classical or quantum). Is it correct? • That can only be confirmed by checking theoreticalpredictions & comparing those to experimental observation! Physics is an experimental science!! Sometimes, this postulateis called The Fundamental Postulate of (Equilibrium) Statistical Mechanics!

  15. 4. Probability Calculations • Finally, we can do some calculations! • Once we have the Fundamental Postulate, we can use Probability Theory to predict the outcome of experiments. • Now, we will go through steps 1., 2., 3., 4. again in detail!

  16. Statistical Formulation of the Mechanical Problem Specification of the System State ≡Microstate • Consider any system of particles. We know that the particles will obey the laws of Quantum Mechanics(we’ll discuss the Classical description shortly). • We’ll emphasize the Quantum treatment.

  17. Consider any system of particles. • Using the Quantum treatment, consider a system with f degrees of freedom can be described by a (many particle!) wavefunction Ψ(q1,q2,….qf,t), where q1,q2,….qf ≡ Set of f generalized coordinates required to characterize the system (needn’t be position coordinates!)

  18. For a system with f degrees of freedom, the many particle wavefunction is formally: Ψ(q1,q2,….qf,t), q1,q2,….qf ≡a set of f generalized coordinates which are required to characterize the system. • A particular quantum state (macrostate) of the system is specified by giving values of some set of f quantum numbers. If we specify Ψat a given time t, we can (in principle) calculate it at any later time by solving the appropriate Schrödinger Equation

  19. Now, briefly look at some simple examples, which might also review some elementary Quantum Mechanics.

  20. Example 1 • Single particle, fixed in position, intrinsic spin s = ½ • Intrinsic angular momentum = ½ћ. • In the Quantum Description of this system, the state of the particle is specified by specifying the projection m of this spin along a fixed axis (which we usually call the z-axis). • The quantum number m can thus have 2 values: ½ (“spin up”) or -½ (“spin down”) So, there are 2 possible states of the system.

  21. Example 2 • N particles (non-interacting), fixed in position. Each has intrinsic spin ½ soEACHparticle’s quantum number mi (i = 1,2,…N) can have one of the 2 values ½. Suppose that N is HUGE:N ~ 1024. • The state of this system is then specified by specifying the values of EACH of the quantum numbers: m1,m2, .. mN.  There are(2)Nuniquestates of the system! WithN ~ 1024, this number isHUGE!!!

  22. Example 3 • A system with N = 3 Particles, fixed in position, each with spin = ½  Each spin is either “up” (↑, m = ½) or “down” (↓, m = -½). • Each particle has a vector magnetic moment μ. • The projection of μalong a “z-axis” is either: μz = μ, for spin “up” or μz = -μ, for spin “down”

  23. Possible States of a 3 Spin System • Given that we know no other information about • this system, all we can say about it is that • It has Equal Probability of Being Found • in Any One of These 8 States.

  24. Put this system into an External Magnetic Field H. • Classical E&M tells us that a particle with magnetic moment μin an external field H has energy: ε = - μH • Combine this with the Quantum Mechanical result: This tells us that each particle has 2 possible energies: ε+ ≡ - μHfor spin “up” ε- ≡ μHfor spin “down” So, for 3 particles, the Stateof the system is specified by specifying each m =   There are(2)3 = 8 Possible States!!

  25. However, if (as is often the case in real problems) we have a partial knowledge of the system (say, from experiment), then, we know that The system can be only in any one of the states which are COMPATIBLE with our knowledge. (That is, it can only be in one of it’saccessible states) “States Accessible to the System” ≡ those states which are compatible with all of the knowledge we have about the system. Its important to use all of the information that we have about the system!

  26. Example 4 • For our 3 spin system, suppose that we measure the total system energy & we find E ≡ - μH • This additional information limits the states which are accessible to the system. Clearly, from the table, • Out of the 8 states, only 3 are • compatible with this knowledge. •  The system must be in one of the 3 states: • (+,+,-) (+,-,+) (-,+,+)

  27. Example 5 • The system is a quantum mechanical, one-dimensional, simple harmonic oscillator, with position coordinate x & classical frequency ω. So the Quantum Energy of this system is: En = ћω(n + ½), (n = 0,1,2,3,….). • The quantum states of this oscillator are then specified by specifying the quantum number n. So, there are essentially an  NUMBER of such states!

  28. Example 6 • The system is N quantum mechanical, one-dimensional, simple harmonic oscillators, at positions xi, with classical frequencies ωi (i = 1,2,.. N). • The Quantum Energies of each particle in this system are: Ei = ћωi(ni + ½), (ni = 0,1,2,3,….). • The system’s quantum states are specified by specifying the values of eachquantum number ni. Here also, there are essentially an NUMBER of such states. • But, there are also a larger number of these than in Example 3!

  29. What about the Classical Description of the state of a many particle System? The Quantum Description is always correct! • But, it is often useful & convenient to make the Classical Approximation. How do we specify the state of the Classical system?

  30. Start with a very simple case: A Single Particle in 1 Dimension: • In classical mechanics, it can be completely described in terms of it’s generalized coordinate q & it’s momentum p. The usual case: consider the Hamiltonian Formulation of classical mechanics, where we talk of generalized coordinates q & generalized momenta p, rather than the Lagrangian Formulation, where we talk of coordinates q & velocities (dq/dt).

  31. The particle obeys Newton’s 2nd Law under the action of the forces on it. • Equivalently, it obeys Hamilton’s Equations of Motion. • q & p completely describe the particle classically. Given q, p at any initial time (say, t = 0), they can be determined at any other time t by integrating the equations of motion.

  32. q & p completely describe the particle classically. Given q, p at any initial time (say, t = 0), they can be determined at any other time t by integrating the Newton’s 2nd Law Equations of Motion forward in time.  Knowing q & p at t = 0 in principle allows us to know them for all time t. q & pcompletely describe the particle for all time. This situation can be abstractly represented in a geometric way discussed on the next page.

  33. At any time t, stating the (q, p) of the particle describes it’s • “State” • Specification of the • “State of the Particle” • is done by stating which point in this plane the particle “occupies”. • Consider the (abstract) 2-dimensional space defined by q, p:≡ “Classical Phase Space”of the particle. • Of course, as q & p change in time, according to the • equation of motion, the point representing the particle • “State”moves in the plane.

  34. To do this, it is convenient to • subdivide the ranges of q & p • into very small rectangles of size: • qp. • q, p are continuous variables, so an  number of points are in this 2-Dimensional Classical“Phase Space” • We’d like to describe the particle “State” classically in a way that the number of states is countable. • Think of this 2-d phase space as divided into small cells of equal area: qp ≡ ho • ho ≡ a small (arbitrary) constant with units of angular momentum .

  35. The 2-d phase space has a large number cells of area: qp = ho. • The (classical) particle “State” is specified by stating which cell in phase space the q, p of the particle is in. Or, by stating that it’s coordinate lies between q & q + q & that it’s momentum lies between p & p + p. • “State” • The phase space cell • labeled by the (q,p) that • the particle “occupies”.

  36. This involves the “small” parameter ho, which is arbitrary. As a side note, however, we can use Quantum Mechanics & the Heisenberg Uncertainty Principle: “It is impossible toSIMULTANEOUSLY specify a particle’s position & momentum to a greater accuracy thanqp ≥ ½ћ” • So, the minimum value ofhois clearly ½ћ. As ho½ћ, • The classical description of the State approaches the quantum description & becomes more & more accurate.

  37. Now!! Lets generalize all of this to a MANY PARTICLE SYSTEM • 1 particle in 1 dimension means we have to deal with a 2-dimensionalphase space. • The generalization to N particles is straightforward, but requires thinking in terms of a very abstract Multidimensionalphase space. • Consider a system with f degrees of freedom:  The system is described classically by f generalized coordinates:q1,q2,q3, …qf f generalized momenta:p1,p2,p3, …pf.

  38. A complete description of the classical “State” of the system requires the specification of: f generalized coordinates: q1,q2,q3, …qf. & f generalized momenta: p1,p2,p3, …pf (N particles, 3-dimensions f = 3N !!)

  39. A complete description of the classical “State” of the system requires the specification of: f generalized coordinates: q1,q2,q3, …qf. & f generalized momenta: p1,p2,p3, …pf (N particles, 3-dimensions f = 3N !!) • So, now lets think VERYabstractly in terms of a 2f-dimensional phase space • The system’s f generalized coordinates: q1,q2,q3, …qf. & f generalized momenta: p1,p2,p3, …pf are regarded as a point in the 2f-dimensional phase spaceof the system.

  40. 2f-dimensional Phase Space: fq’s & fp’s: • Each q & each p label an axis (analogous to the 2-d phase space for 1 particle in 1 dimension). • Subdivide this phase space into small “cells” of 2f-dimensional“differential volume”: q1q2q3…qfp1p2p3…p1f≡ (ho)f • The classical “State” of the system is then ≡ the cell in this 2f-dimensional phase space that the system “occupies”.

  41. Reif, as all modern texts, takes the viewpoint that the system’s “State” is described by a 2f-dimensional phase space ≡ “The Gibbs Viewpoint” The system “State”≡ The cell in this phase space that the system “occupies”. • Older texts take a different viewpoint ≡ “The Boltzmann Viewpoint”: In this viewpoint, each particle moves in it’s own 6-dimensional phase space • In this view, specifying the system “State” requires specifying each cell in this phase space that each particle in the system “occupies”.

  42. Summary Specification of the System State: In Quantum Mechanics: • Enumerate & label all possible system quantum states. In Classical Mechanics: • Specify which cell in 2f-dimensional phase space The system is in. • Need the coordinates & momenta of all particles & the “box” in the p-q plane the system occupies. As ho→ ½ћ, the classical & quantum descriptions become the same.

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