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Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel Physics 452 Homework First homework assignment: Tuesday Jan 10 by 10pm Assignment # 1: Problems 5.22, 5.23, 5.24 in the textbook Second homework assignment: Thursday Jan 12 by 10pm Phys 452 Quantum statistical mechanics N particles ( N 1) Thermal equilibrium, T Quantization of the energy for individual particles ( E1 , E2 , E3 ,...) Total energy: E N1E1 N 2 E2 N3 E3 ... How many ways we get the configuration Q N1 , N2 , N3 ,... N , N 1 2, N 3, ... ? Phys 452 Quantum statistical mechanics Example: 3 –particle system • Textbook example Infinite square well E 2 2 ma 2 na2 nb2 nc2 • In-class example/ pb 5.23 Harmonic oscillator 3 E na nb nc 2 For each type of particles: • List all the possible configurations • Determine the number of combinations of each configuration • Determine the probability of each configuration for a given energy Phys 452 Quiz 1b Consider 3 distinguishable particles in a harmonic oscillator potential. If the total energy of the system is E 9 / 2 how many possibilities there are to get the configuration (1,1,1,0,…)? A. 10 B. 4 C. 1 D. 6 E. 2 Phys 452 Quiz 1c Consider 3 fermions particles in a harmonic oscillator potential. If the total energy of the system is E 9 / 2 how many possibilities there are to get the configuration (1,1,1,0,…)? A. 10 B. 4 C. 1 D. 6 E. 2 Phys 452 Quiz 1d Consider 3 bosons in a harmonic oscillator potential. If the total energy of the system is E 9 / 2 how many possibilities there are to get the configuration (1,1,1,0,…)? A. 10 B. 4 C. 1 D. 6 E. 2 Phys 452 Quantum statistical mechanics Statistical configuration number: • Distinguishable particle • Identical fermions d nNn Qdis N1 , N 2 , N3 ,.. N ! n 1 N n ! Q fermions N1 , N 2 , N 3 ,.. n 1 • Identical bosons dn ! N n ! d n N n ! Qbosons N1 , N 2 , N3 ,.. n 1 N n dn 1! N n ! d n 1! Work out example: harmonic oscillator, infinite square well Phys 452 Quiz 2a Let’s consider the Carbon atom: with 6 electrons To be distributed in the energy levels E1 and E2 What is the number of combinations Q(2, 4, 0,....) ? A. 6! B. 15 C. 8! D. 70 E. 45 Phys 452 Quiz 2b Let’s consider the Carbon atom: with 6 electrons to be distibuted in the shells (1s)(2s)(2p) What is the number of combinations Q(2, 2, 2, 0,....) ? A. 6! B. 15 C. 8! D. 70 E. 45 Phys 452 Quantum statistical mechanics The most probable configuration Lagrange multipliers, using: N Nn n 1 E N n En n 1 G N1 , N 2 , N3 ,... ln Q N N n E N n En n 1 n 1 Maximizing Q: G 0 N n Expressing Nn in terms of and Phys 452 Quantum statistical mechanics Most probable occupation number: • Distinguishable particle • Identical fermions • Identical bosons Nn dne Nn Nn En dn En e e 1 dn 1 En 1 Phys 452 Quantum statistical mechanics Significance of and : En e E k BT dimensionless related to the temperature related to chemical potential 1 k BT (T ) k BT Phys 452 Quantum statistical mechanics Calculation of and : Case of ideal gas (free electron gas) One spherical shell kz kF “degeneracy” Fermi surface kx density of states d shell dk ky Vk 2 dk 2 2 N Bravais k-space Nk dk k 0 Volume in k-space of each individual state Vunit 3 V E k 0 N k Ek d k Phys 452 Quantum statistical mechanics Calculation of and : Case of ideal gas : distinguishable particles N Nk dk k 0 N k Ek d k k 0 Then using 2 E E m N eV 2 3/2 1 k BT 3 m E Ve 2 2 We can express 2 3/2 (T ) kBT 3N 2 Phys 452 Quantum statistical mechanics Case of ideal gas : distinguishable particles N 3 2 2 (T ) k BT ln ln V 2 mk BT 3 E Nk BT 2 Phys 452 Quantum statistical mechanics Most probable occupation number: E / k BT Maxwell-Boltzmann n E e • Distinguishable particle statistic • Identical fermions nE • Identical bosons nE 1 E / k BT e 1 1 e E / kBT 1 Fermi- Dirac statistic Bose-Einstein statistic Phys 452 Quiz 2c What is the maximum possible value for the density of occupation in case of fermions? A. B. 1/ e 1 C. 1 D. 0 E. undetermined Phys 452 Quantum statistical mechanics Fermi-Dirac distribution: nE 1 E / k BT e 1 (0) EF nE 1 e E EF / kBT 1 T 0 n 1 if E 0 n 0 if E 0