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.
1 Uncertainty defined 1 . 2 The Uncertainty Principle 3 . ... all we have the Hamiltonian operator, and its uncertainty ΔH is a perfect candidate for the ‘energy ... a real number used to describe the way systems change. Unless we define Δt in a precise way we cannot hope for a well-defined uncertainty relation.
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