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HW10_Solutions - PHYS851 Quantum Mechanics I Fall 2008...

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PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 10 1. [10 pts] Prove that | α ) = D ( α ) | 0 ) , where | α ) is the Glauber coherent state, | 0 ) is the harmonic oscillator ground state, and D ( α ) = e αA - α * A , is the so-called ‘displacement operator’. Here A and A are the harmonic oscillator creation and annihilation operators satisfying [ A, A ] = 1. You may need the Zassenhaus formula e B + C = e B e C e - [ B,C ] / 2 , which is valid for any two operators B and C provided [ B, [ B, C ]] = [ C, [ B, C ]] = 0. Answer: Let B = αA and C = α * A . Then [ B, C ] = −| α | 2 [ A , A ] = | α | 2 , which commutes with everything, so we can use the Zassenhaus formula, which gives e αA - α * A = e αA e - α * A e -| α | 2 / 2 Now we have e - α * A | 0 ) = summationdisplay n =0 ( α * ) 2 n ! A n | 0 ) = | 0 ) because A n | 0 ) = 0 unless n = 0. So we end up with D ( α ) | 0 ) = e -| α | 2 / 2 e αA | 0 ) = e -| α | 2 / 2 summationdisplay n =0 α n n ! ( A ) n | 0 ) Now recall that | n ) = ( A ) n n ! | 0 ) which gives D ( α ) | 0 ) = e -| α | 2 / 2 summationdisplay n =0 α n n ! | n ) = | α ) 1
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2. [15 pts]The Boltzman formula for the mean energy at thermal equilibrium is derived from maximizing the entropy, and gives E avg = n E n e - βE n /Z , where β = ( K b T ) - 1 , with K b being the Boltzman constant and T being the temperature. The partition function is Z = n e - βE n . Here the sum is over all allowed states of the system and E n is the energy of the n th state. Show that E avg = ∂β ln Z . Then compute the sum analytically, and use this formula to compute the average energy, ( E ) , of a quantum harmonic oscillator at temperature T . Answer: ∂β ln Z = 1 Z ∂β Z = P n E n e - βEn Z = E avg Let E n = planckover2pi1 ω ( n + 1 / 2).
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