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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) barb2right 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 barb2right 1 state for Bosons barb2right 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 μ 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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