PHY4221Fall05_Notes3x28Addendumx29 - PHY4221 Quantum...

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PHY4221 Quantum Mechanics I Fall 2004 In the study of a quantum simple harmonic oscillator, we use the factorization method 1 to introduce the creation and annihilation operators, i.e. ˆ a and ˆ a . These two operators provide an elegant algebraic tool to examine the properties of the system, in particular determining the energy eigenstates and eigenvalues. The energy eigenstates, known as the number states , form a complete orthonormal set of basis vectors to describe the simple harmonic oscillator. In addition to this number state basis, there yet exists another extremely useful basis, namely the Glauber coherent states . These Glauber coherent states are closely related to the Hilbert space of entire analytic functions developed by Bargmann, which is commonly known as the Fock-Bargmann space . 2 The Glauber coherent states are the eigenstates of the non-Hermitian annihilation operator a , i.e. a | z i = z | z i , (1) where the eigenvalue z is complex. It is easy to verify that these states have the form | z i = exp za - z * a · | 0 i = exp ˆ - | z | 2 2 ! exp za · | 0 i = exp ˆ - | z | 2 2 ! X
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This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

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PHY4221Fall05_Notes3x28Addendumx29 - PHY4221 Quantum...

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