Introduction to Quantum Mechanics · Problem 6.31

Symmetries and Conservation Laws — David J. Griffiths

Physics — Quantum Mechanics · translation_operator_spectral

Problem

In deriving Equation 6.3 we assumed that our function had a Taylor series. The result holds more generally if we define the exponential of an operator by its spectral decomposition, T^(a)=∫e−iap/ℏ ∣p⟩⟨p∣ dp,(6.80)\hat{T}(a) = \int e^{-iap/\hbar}\,|p\rangle\langle p|\,dp, \tag{6.80} rather than its power series. Here I've given the operator in Dirac notation; acting on a position-space function (see the discussion on page 123) this means T^(a) ψ(x)=∫−∞∞e−iap/ℏ fp(x) Φ(p) dp,(6.81)\hat{T}(a)\,\psi(x) = \int_{-\infty}^{\infty} e^{-iap/\hbar}\,f_p(x)\,\Phi(p)\,dp, \tag{6.81} where Φ(p)\Phi(p) is the momentum space wave function corresponding to ψ(x)\psi(x) and fp(x)f_p(x) is defined in Equation 3.32. Show that the operator T^(a)\hat{T}(a), as given by Equation 6.81, applied to the function ψ(x)=λ e−λ∣x∣\psi(x) = \sqrt{\lambda}\,e^{-\lambda|x|} (whose first derivative is undefined at x=0x = 0) gives the correct result.

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