# 348-problem21 - x p Φ Now con-sider the eigenvalue problem...

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Chem. 540 Instructor: Nancy Makri PROBLEM 21 We have postulated the form of the momentum operator in the position representation and found the expression of momentum eigenfunctions in position space. Let’s consider the reverse problem. Consider the ket x , which is an eigenstate of the position operator, and its “wavefunc- tion” in momentum space ( ) x p x p ≡ Φ . Use the known bra-ket properties and the obtained expression for x p to find
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Unformatted text preview: ( ) x p Φ . Now con-sider the eigenvalue problem for position states in the momentum representation, ˆ ( ) ( ) x x x p x p Φ = Φ . Show that this eigenvalue relation is satisfied if the position operator has the form ˆ x i p ∂ = ∂ h . Further, by operating on an arbitrary wavefunction ( ) p ψ , show that this choice produces the fa-miliar result for the position-momentum commutator, ˆ ˆ [ , ] x p i = h ....
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