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# 09_580_hw_5 - Physics 580 Quantum Mechanics I Prof P M...

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Physics 580 Handout 9 28 September 2010 Quantum Mechanics I webusers.physics.illinois.edu/ goldbart/ Prof. P. M. Goldbart 3135 (& 2115) ESB University of Illinois 1) Exponential operators : In this question we shall consider a function of the two oper- ators Λ and Ω. First, we shall specialise to the case in which the operators are such that their commutator is proportional to the identity, [Λ , Ω] = d I. Often you will see this writ- ten [Λ , Ω] = d ; a situation described by the statement ‘the commutator is a c-number (or commuting or classical number)’. By following the strategy below, we will prove that for operators whose commutator is a c-number: exp Λ exp Ω = exp ( Λ + Ω + 1 2 , Ω] ) . a) Check the consistency of this equation by expanding the exponential functions retaining terms of quadratic and lower order in the operators Λ and Ω. Is the theorem consistent to this order? b) Introduce the c-number µ and the operator-valued functions f ( µ ) e µ Λ e µ , g ( µ ) e µ (Λ+Ω)+ 1 2 µ 2 , Ω] . Evaluate df/dµ (remembering that ordering of operator matters). c) Evaluate dg/dµ . d) By expanding exp ( µ Ω), and hence evaluating [Λ , exp ( µ Ω)], prove that f and g satisfy the same differential equation, viz. , dh = h ( µ ) ( µd + Λ + Ω) . e) Is f (0) equal to g (0)? Hence prove that f ( µ ) = g ( µ ). By setting µ = 1 we have the desired result.

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09_580_hw_5 - Physics 580 Quantum Mechanics I Prof P M...

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