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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 10 1. [10 pts] Prove that | ) = D ( ) | ) , where | ) is the Glauber coherent state, | ) is the harmonic oscillator ground state, and D ( ) = e A - * 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 = A and C = * A . Then [ B,C ] = | | 2 [ A ,A ] = | | 2 , which commutes with everything, so we can use the Zassenhaus formula, which gives e A - * A = e A e- * A e-| | 2 / 2 Now we have e- * A | ) = summationdisplay n =0 ( * ) 2 n ! A n | ) = | ) because A n | ) = 0 unless n = 0. So we end up with D ( ) | ) = e-| | 2 / 2 e A | ) = e-| | 2 / 2 summationdisplay n =0 n n ! ( A ) n | ) Now recall that | n ) = ( A ) n n ! | ) which gives D ( ) | ) = e-| | 2 / 2 summationdisplay n =0 n n ! | n ) = | ) 1 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- E n /Z , where = ( K b T )- 1 , with K b being the Boltzman constant and T being the temperature. The partition function is Z = n e- E 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 = ln Z ....
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