PHYS_342_hmwrk5_spin_statsitics

PHYS_342_hmwrk5_spin_statsitics - PHYS342 FallSemester2011...

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1 PHYS 342 Fall Semester 2011 Homework No. 5 Due: October 28, 2011 1. Suppose there are two identical noninteracting quantum particles with the SAME SPINS are located at positions x 1 and x 2 . The two particles each satisfy Schrödinger’s Equation 22 11 111 2 1 222 2 2 () 2 2 x Ex mx x  This implies that there are two available quantum states for the system as shown below. Schrödinger’s Equation for the system becomes  2 12 1 2 (; ) 2 x xE x mx x      where x x is the multi-particle wave function. The questions below are designed to make you think about the possible ways the two particles can be distributed between these two energy levels? E 1 E 2 Ψ 1 (x 1 ) Ψ 2 (x 2 )
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2 a) If the two particles are bosons , complete the table below to show the three possible ways to arrange the two particles between the two energy states. Indicate whether a state is occupied by placing a dot on each energy level . What is the total energy and system wave function for each case? Make sure the wavefunctions you have written is symmetric under particle exchange? Particle occupation (bosons) 12 (; ) xx  Total Energy= b) If the two particles are fermions , complete the table below to show the only way to arrange the two particles between the two energy states. Indicate whether a state is occupied by placing a dot on each energy level . What is the total energy and system wave function? Make sure the wavefunction you have written is antisymmetric under particle exchange! Particle occupation (fermions) Total Energy= E 1 E 2 Ψ (x 1 ) Ψ (x 2 ) E 1 E 2 Ψ 1 (x 1 ) Ψ 2 (x 2 ) E 1 E 2 Ψ 1 (x 1 ) Ψ 2 (x 2 ) E 1 E 2 Ψ 1 (x 1 ) Ψ 2 (x 2 )
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3 2. A completely general way to write down an antisymmetric wavefunction for a many particle fermion system is to use a technique called Slater determinants. Suppose we have three fermions (electrons) at positions x 1 , x 2 and x 3 . Each electron independently satisfies Schrödinger’s Equation, so we have The unique set of quantum numbers for each eigenfunction solution is indicated by the numbered subscript “1” , “2” or “3” associated with each wave function.
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PHYS_342_hmwrk5_spin_statsitics - PHYS342 FallSemester2011...

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