HW10_Solutions - PHYS851 Quantum Mechanics I Fall 2008...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 10 1. [10 pts] Prove that | α ) = D ( α ) | ) , where | α ) is the Glauber coherent state, | ) 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 | ) = ∞ summationdisplay n =0 ( − α * ) 2 n ! A n | ) = | ) because A n | ) = 0 unless n = 0. So we end up with D ( α ) | ) = e-| α | 2 / 2 e αA † | ) = e-| α | 2 / 2 ∞ summationdisplay n =0 α n n ! ( A † ) n | ) Now recall that | n ) = ( A † ) n √ n ! | ) which gives D ( α ) | ) = e-| α | 2 / 2 ∞ summationdisplay n =0 α n √ n ! | n ) = | α ) 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- 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 ....
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HW10_Solutions - PHYS851 Quantum Mechanics I Fall 2008...

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