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Unformatted text preview: CHEM 356, Lecture 8, Fall 2009 1 General Quantum–Mechanical Operators . Quantities, such as the translational kinetic energy , linear momentum, total energy, are known as dynamical variables . Only when a dynamical variable is constant in a stationary state can it be precisely determined, so that the only possible EXACT values which may be observed for dynamical variables, such as the translational (kinetic) energy, , the ? –component of the linear momentum, ? , and the total energy, , are those given by the corresponding eigenvalue equations ˆ = ; ˆ ? = ? ; ℋ = . Dynamical variables that can be measured experimentally are called observables . These concepts are embodied in Postulate 3. (a) For every dynamical variable there is a corresponding linear operator . If the dynamical variable is capable of exact experimental determination, its only possible exact values are those given by the eigenvalues of the equation Ψ = Ψ . (b) For a conservative system, we have = . Construction of New Operators : usually a two–step process, namely: i) Write down the classical expression as a function of the coordinate positions, time, linear momenta for the dynamical variable corresponding to the desired operator, then ii) substitute the required operator forms for these quantities. CHEM 356, Lecture 8, Fall 2009 2 Example : angular momentum about the ? –axis. Classically, this is given by ( r × p ) ? = ? ? − ? ? = ? . Hence we obtain the operator Significance of in Conservative Systems. Classically: for stationary classical waves is such that ∣ ∣ 2 = ∗ gives a measure of the relative energy density at any given position. Quantum mechanically: observables which we measure experimentally for any particle (or system of par ticles) are real–valued, and any final result corresponding to an observation must be a real result. For a conservative system, we have for the total energy the expression ˆ + ˆ = , and, upon multiplying each side of this equation by ∗ , we obtain ∗ ˆ + ∗ ˆ = ∗ , which we can rewrite as ∗ ˆ + ( r ) ∗ = ∗ CHEM 356, Lecture 8, Fall 2009 3 Probabilistic interpretation: The differential probability, the probability density and the differential volume element...
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 Fall '09
 Prof.Iaskjd
 Energy, Kinetic Energy, expectation value, translational kinetic energy, Conservative Systems

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