09statistics1 - Quantum Statistics Determine probability...

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P461 - Intro. Quan. Stats. 1 Quantum Statistics • Determine probability for object/particle in a group of similar particles to have a given energy • derive via: a. look at all possible states b. assign each allowed state equal probability c. conserve energy d. particles indistinguishable (use classical - distinguishable - if wavefunctions do not overlap) e. Pauli exclusion for 1/2 integer spin Fermions • classical can distinguish and so these different: but if wavefunctions overlap, can’t tell “1” from “2” from “3” and so the same state 1 3 6 3 2 2 1 3 3 2 2 1 1 = = = = = = = E E E E E E E total
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P461 - Intro. Quan. Stats. 2 Simple Example • Assume 5 particles with 7 energy states (0,1,2,3,4,5,6) and total energy = 6 • Find probability to be in each energy state for: a. Classical (where can tell each particle from each other and there is no Pauli exclusion) b. Fermion (Pauli exclusion and indistinguishable) c. Boson (no Pauli exclusion and indistinguishable) b different ways to fill up energy levels (called microstates) 1 E=6 plus 4 E=0 1 E=5 + 1 E=1 + 3 E=0 1 E=4 + 1 E=2 + 3E=0 * 1 E=4 + 2E=1 + 2E=0 * 1E=3 + 1E=2 + 1E=1 + 2E=0 2E=3 + 3E=0 1E=3 + 3E=1 + 1E=0 3E=2 + 2E=0 * 2E=2 + 2E=1 + 1E=0 1E=2 + 4E=1 can eliminate some for (b. Fermion) as do not obey Paili exclusion. Assume s=1/2 and so two particles are allowed to share an energy state. Only those * are allowed in that case
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P461 - Intro. Quan. Stats. 3 Simple Example • If Boson or classical particle then can have more then 1 particle in same state….all allowed • But classical can tell 1 particle from another State 1 E=6 + 4 E=0 b 1 state for Bosons b 5 states for Classical (distinguishable) assume particles a,b,c,d,e then have each of them in E=6 energy level • for Classical, each “energy level” combination is weighted by a combinatorial factor giving the different ways it can be formed 5 1 1 1 ! 4 1 ! 5 ! ! ! ! ! ! 5 4 3 2 1 = = = N N N N N N W
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P461 - Intro. Quan. Stats. 4 Simple Example • Then sum over all the microstates to get the number of times a particle has a given energy for Classical, include the combinatoric weight (W=1 for Boson or Fermion) Energy Probability: Classical Boson Fermion 0 .4 .42 .33 1 .27 .26 .33 2 .17 .16 .20 3 .095 .08 .07 4 .048 .04 .07 5 .019 .02 0 6 .005 .02 0 1050 210 , 5 , 10 : 15 3 , 5 , 3 : 50 10 , 5 , 10 : ) ( Pr = = = = = = = W N states Classical W N states Fermion W N states Boson W N W N E ob E μ
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P461 - Intro. Quan. Stats. 5 Distribution Functions • Extrapolate this to large N and continuous E to get the probability a particle has a given energy Probability = P(E) = g(E)*n(E) g=density of states = D(E) = N(E) n(E) = probability per state = f(E)=distribution fcn.
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This note was uploaded on 10/02/2009 for the course PHYS 460 taught by Professor Johnson,c during the Spring '08 term at Northern Illinois University.

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09statistics1 - Quantum Statistics Determine probability...

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