Use the operator method of Chapter 7 to derive the free-particle propagator with

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch The path integral

Problem

Use the operator method of Chapter 7 to derive the free-particle propagator K(x,t;x′,0)=⟨x∣e−iH^t/ℏ∣x′⟩K(x,t;x',0)=\langle x|e^{-i\hat Ht/\hbar}|x'\rangle with H^=p^2/2m\hat H=\hat p^2/2m, by inserting a resolution of the identity in the momentum basis and carrying out the resulting Gaussian integral. Then evaluate the result for an electron (m=9.11×10−31 kgm=9.11\times10^{-31}\,\mathrm{kg}) with t=1.0×10−15 st=1.0\times10^{-15}\,\mathrm{s} and x−x′=1.0×10−9 mx-x'=1.0\times10^{-9}\,\mathrm{m}. ((a)) Give the magnitude ∣K∣|K|. ((b)) Give the phase Scl/ℏS_{\rm cl}/\hbar (in radians). ((c)) Show that your propagator is Ffree(t) eiScl/ℏF_{\rm free}(t)\,e^{iS_{\rm cl}/\hbar} of the quadratic path integral, naming FfreeF_{\rm free} and SclS_{\rm cl}.

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