15_580_hw_11 - Physics 580 Quantum Mechanics I Prof P M...

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Physics 580 Handout 15 9 November 2010 Quantum Mechanics I webusers.physics.illinois.edu/ goldbart/ Homework 11 Prof. P. M. Goldbart University of Illinois 1) Coherent states : In quantum optics, the classical limit of quantum mechanics, quantum field theory and quantum statistical mechanics, there is a very important set of states, known as coherent states , which may be constructed from the harmonic oscillator energy eigenstates {| n ⟩} . Each coherent state is labelled by a single complex number z ; it is denoted | z , and defined by | z ⟩ ≡ exp( za ) | 0 , where the ket | 0 is the harmonic oscillator ground state. a) Show that the coherent state | z is an eigenstate of the annihilation operator a . What is its eigenvalue? b) Evaluate z | a in terms of z | and z . c) Show that ( d/dz ) | z = a | z . d) Show that the inner product of two coherent states is given by z | z = exp( z z ) . Consider the function of two complex variables f ( z, z ). The operator obtained by inserting a for z and a for z is generally ambiguous because the order of the operators is not specified. For any f we can unambiguously define an operator f ( a , a ), in which the operator ordering ambiguity is resolved by the prescription that, in any product, all annihilation operators lie to the right of all creation operators. Operators of this form are called normal ordered operators. e) Show that for normal ordered operators we have
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15_580_hw_11 - Physics 580 Quantum Mechanics I Prof P M...

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