STATES OF A MODEL SYSTEM the systems we are interested in has many available quantum states - many states can have identical energy --> multiplicity (degeneracy) of a level: number of quantum states with the same energy - it is the number of quantum states that is important in thermal physics, not the number of energy levels! Examples for quantum states and energy levels of several atomic systems: (multiplicity for each energy level shown in the brackets) 1. Hydrogen (one electron + one proton) 2. Lithium (3 electrons + 3 protons neutrons) 3. Boron (five electrons + 5 protons neutrons) 4. Particle confined to a cube n x, n y, n z --> quantum numbers : 1, 2, 3, …k,... Quantum states of one particle systems --> orbitals

Binary model systems - elementary magnets pointing up or down - cars in a parking lot - binary alloys -m magnetic moment+m magnetic moment Occupied or type A atom Unoccupied or type B atom A single state of the system: All states of the system generated by:-->generating function Total number of states: 2 N ; N+1 possible values of the total magnetic moment: M=Nm, (N-2)m, (N-4)m,...-Nm number of states >> possible values of total magnetic moments (if N>>1) if the magnetic moments are not interacting M will determine the E total energy of the system in a magnetic field! ( number of states >> possible energy values) --> some states have large multiplicity

Enumeration of States and the Multiplicity Function (Let us assume N even) N  : number of up spins, N  : number of down spins spin excess Multiplicity function g(N,s) of a state with a given spin excess Ex. Form of g(10,s) as a function of 2s: Binary alloy systems: (N-t) A atoms and (t) B atoms on N sitesthe same multiplicity function

Sharpness of the Multiplicity Function - g(N,s) is very sharply peaked around s=0; - we want get a more analytical form of g(N,s) when N>>1 and s<<N - we will follow the same procedure as for the random-walk problem! - we use the Stirling approximation: and after find: Width of the g(N,s) multiplicity function governed by for s/N=(1/2N) 1/2 the value of g is e -1 of g(N,0) g(N,s) is a Gaussian-like distribution! For N>>1 the distribution gets very sharp --> strong consequences for thermodynamic systems

Problems 1. Prove that: 2. Prove the Stirling approximation: 3. Approximate in the limit of large energy values the  (  ) density of states for a particle confined in a 3D box where n( ,  +d  ) represent the number of states with energy between  and  +d . 4. Starting from the multiplicity function for a binary model system, approximate the number of possible states of the system, when N=100 and and s is between 0 and +10.

Extra problem 1. (**) Using the entropy formula given by Renyi calculate the entropy of a binary model system presuming that all microstates are equally probable. (In the above Renyi formula the summation is over all possible microstates, and P i represents the probability of microstate i)