MIT 6 453 - Problem Set 4 (6 pages)

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Problem Set 4



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Problem Set 4

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Lecture Notes


Pages:
6
School:
Massachusetts Institute of Technology
Course:
6 453 - Quantum Optical Communication

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MIT OpenCourseWare http ocw mit edu 6 453 Quantum Optical Communication Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6 453 Quantum Optical Communication Problem Set 4 Fall 2008 Issued Thursday September 25 2008 Due Tuesday October 7 2008 Problem 4 1 Here we shall show that the creation operator a does not have any non zero eigen kets Suppose that a non zero ket satis es a 1 where is a complex number Use the completeness of the number kets to expand as follows bn n n 0 where bn n Substitute this expansion into Eq 1 and show that the only possible solution is bn 0 for all n i e the creation operator has no non zero eigenkets Problem 4 2 Here we shall work out some properties of the coherent states Let a and a be the annihilation and creation operators for the frequency quantum harmonic oscillator discussed in class Let C be the coherent states n exp 2 2 n n n 0 where n 0 n are the number states and C is an arbitrary complex number a Use the orthonormality of the number states and the power series for the ex ponential function to evaluate the inner product between two coherent states and Are the coherent states normalized to unit length Are coherent states with di erent eigenvalues orthogonal b Use the completeness of the number states to show that the coherent states are overcomplete i e 2 d I where d2 d 1 d 2 with 1 Re and 2 Im and the integration region is the entire complex plane 1 c Use the result from b to show that 2 d a a I 2 d a Ia 2 d 2 a a a Ia 2 d a a a a a a 2 1 Problem 4 3 Here we shall develop a little commutator calculus that will be needed in the next problem Let a and a be the annihilation and creation operators respectively of a quantum harmonic oscillator and let a 1 Re a and a 2 Im a be the associated quadrature operators i e the normalized versions of position and momentum for a mechanical



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