Question

5. Coherent States (Answer only question 5 for part a, b, and c)

A coherent state is an Eigenstate of annihilation / lowering operator

c) Baker- Campbell- Hausdorff Formula [Hint: Define the functions fa-eaA+8), ģ(A)-eä eABe-12 č. Note that these functions are equal at -0, and show that they satisfy the same differential equation: df/di (A+ B)f and dg/da (A+B)g Therefore, the functions are themselves equal for all λ.] A useful application of BCH formula is given in problem 5 5.Coherent States As Sal discussed in the last Discussion, a coherent state is an eigenstate of an annihilation/lowering operator. âl2)-λλ), where 12)-eλαf 10) (not normalized) 1 is, in general, a complex number, sincea is a non-hermitian operator. a) Prove that is a normalized coherent state. [Hint: Use the Baker- Campbell- Hausdorff formula in problem 4.] And show that a 12) λ 12) b) Prove the minimum uncertainty relation for such state by showing that the condition is satisfied. [See Griffiths eq. (3.69).] c) Using the Baker- Campbell- Hausdorff formula, show that a coherent state can be obtained (up to a normalization constant) by applying the translation operator, exp(ip xo/h), to the ground state of a harmonic oscillator exp(ip xo/h) l0) where xo is the displacement distance [It is straightforward to see that exp(ip xo/h) is a translation operator since for a function f (x) that can be expanded in a Taylor series, f(x xo)exp(ip xo/h)f(x).]

Answer 1

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