This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 socalled 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 ....
View
Full
Document
 Fall '08
 MichaelMoore
 mechanics, Work

Click to edit the document details