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Unformatted text preview: Introduction to Quantum and Statistical Mechanics Note 11 Statistical Mechanics from a Quantum View Assigned Reading: Hagelstein, Senturia and Orlando, Chaps. 25 and 26, small part of Chap. 27 11.1 Overview When a quantum system has a large number of eigenstates Ω , each of which has various probabilities of occupancy p j for j =1… Ω , in theory we know every physical quantities by counting the contributions from each state. For example, we can know the total number of particles N and total energy E as (notice that we have used N as the number of states in Note 10, which is Ω here): ∑ ∑ = = = = Ω 1 Ω 1 j j j j j E p E p N (11.1) Even when we use DOS g(E) and follow the estimation in Eq. (10.25): ( 29 ( 29 ∫ ∑ ∑ ∞ ∞- 2245 2245 < 2245 < dE E Q E p E g E Q E p Q p Q j j j j j j j j ) ( ) ( ) ( | ˆ | φ φ (11.2) We still need to find a readily accessible way to find p j or p(E) in order to obtain the expectation values of the physically measurable quantities, without resorting to calculations in each eigenstates. This is the realm of statistical mechanics when the numbers of particles and available states are very large. For sure, if we know the wave function for the system as a solution of the full Hamiltonian, then we can use our previous method to determine the amplitude coefficient c j corresponding to the eigenstate j , and p j will then be |c j | 2 . However, for a system with many states and particles, this is prohibitively expensive to compute. 11.2 Equilibrium and Entropy What is the alternative method, which is acceptably accurate though not exact, to estimate p j ? Max Planck made the proposal for this estimate from the Maxwell relations we will introduce below. We will first define “equilibrium”, and then we can estimate how each perturbation will evolve by “particle interactions” or “force coupling” that creates off-diagonal matrix elements to drive the system back to equilibrium. Although the Fermi’s Golden rule plays a major role in determining the transition rate from an initial eigenstate to quasi-continuum final states, we are still in lack of convenient estimation of p j or p(E) to start with. In equilibrium, since ALL transitions, microscopic or macroscopic, need to balance out exactly, for arbitrary two states φ i and φ j in Ω , we will have the transition rate from φ i to φ j ( R f ) to be exactly the same as that φ j to φ i ( R r ). This is called the “detailed balance” in Fig. 11.1, i.e., balance is established for every definable pair of states in equilibrium. 1 Fig. 11.1. Equilibrium defined as detailed balance between the microscopic states. ( Exercise 11.1 ) If a system reaches steady state that all time derivatives are zero, does it necessarily imply equilibrium? Does it necessarily imply that the system has no energy exchange outside the system?...
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This note was uploaded on 11/26/2010 for the course ECE 3060 at Cornell.
- The Land