Transcription of Quantum Physics II, Lecture Notes 5 - MIT OpenCourseWare
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Quantum Physics II, Lecture Notes 5 - MIT OpenCourseWare
UNCERTAINTY PRINCIPLE AND COMPATIBLE OBSERVABLES B. Zwiebach October 21, 2013 Contents 1 Uncertainty defined 1 2 The Uncertainty Principle 3 3 The Energy-Time uncertainty 6 4 Lower bounds for ground state energies 9 5 Diagonalization of Operators 11 6 The Spectral Theorem 12 7 Simultaneous Diagonalization of Hermitian Operators 16 8 Complete Set of Commuting Observables 18 1 Uncertainty defined As we know, observables are associated to Hermitian operators. Given one such operator A we can use it to measure some property of the physical system, as represented by a state . If the state is in an eigenstate of the operator A, we have no uncertainty in the value of the observable, which coincides with the eigenvalue corresponding to the eigenstate. We only have uncertainty in the value of the observable if the physical state is not an eigenstate of A, but rather a superposition of various eigenstates with different eigenvalues.
state Ψ of the quantum system. Let ΔA and ΔB denote the uncertainties of A and B, respectively, in the state Ψ. Then we have \ 1 . 2 (ΔA) 2 (ΔB) 2 . 2i. The left hand side is a real, non-negative number. For this to be consistent inequality, the right-hand side must also be …
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