ExerClass1

# ExerClass1 - PHY4211 Quantum Mechanics I Fall Term of 2005...

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PHY4211 Quantum Mechanics I Fall Term of 2005 Exercise in Page 7 of your Lecture Notes (Chapter 1): Consider the function () ( ) ( ) exp exp f xx A x B = , its first derivative could be obtained as: ( ) () , xA xB xA xB xA xB xA xB x Ax B x B x B x A x B xA xB xA xB df x dd ee e e dx dx dx Ae e e Be Ae e Be e Be e e Be ABee e Be ⎛⎞ =+ ⎜⎟ ⎝⎠ =+−+ ⎡⎤ + ⎣⎦ ; (1) Now consider the term , xA eB : [] {} 00 0 0 1 1 1 1 1 0 11 0 ,, !! ! , ! , ! , ! , ! , 1! , nn n n xA n n n n mn m nm n n mnm n n n n xA x A B x B A B n xAB n x AA BA n x AB A A n x nA n xA n xABe ∞∞ == = = ∞− −− = = ⎢⎥ = ⎧⎫ = ⎨⎬ ⎩⎭ = = = = ∑∑ . (2) Note that in the above calculation we need that n-1>=0 and so n is not smaller than 1. But this is in fact not a problem at all: since if n=0 the corresponding component of exp(xA) is 1 which is a component commuting with B and thus has no contribution to the last result. If you like, you could drop this component at the very first of the whole derivation.

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ExerClass1 - PHY4211 Quantum Mechanics I Fall Term of 2005...

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