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Properties of Hermitian Operators
If A and B are Hermitian, then A + B is Hermitian [A, B] is anti-Hermitian The Symmeterized Sum ½ (AB + BA) is Hermitian if additionally [A, B] = 0, AB is Hermitian If A and B are anti-Hermitian, then iA is Hermitian [A, B] is Hermitian if additionally [A, B] = 0, AB is Hermitian
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Theorems Concerning Hermitian Operators
The eigenvalues of a Hermitian Operator are real. Two eigenfunctions of a Hermitian Operator that correspond to different eigenvalues are orthogonal. Eigenfunctions of a Hermitian Operator that belong to a degenerate eigenvalue can always be chosen to be orthogonal. Commuting Hermitian Operators have simultaneous eigenfunctions. The set of eigenfunctions of any Hermitian Operator (physical observables) form a complete set.
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Postulates of Quantum Mechanics
The “State” of a system is described by a function Y of the coordinates and the time. This function, called the state function or wave function, contains all the information that can be determined about the system. We further postulate that Y is single- valued, continuous, and quadratically integrable. For continuum states, the quadratic integrability requirement is omitted. To every physical observable there corresponds a linear Hermitian operator. To find this operator, write-down the classical mechanical expression for the observable in terms of [cannonical coordinates], and then replace each coordinate x by the operator [multiply by x] and each momentum component p by the operator –I (h-bar) d/dx. The only possible values that can result from measurements of the physically observable property B are the eigenvalues bi in the equation Bgi = bigi, where B is the operator corresponding to the property B. The eigenfunctions gi are required to be well-behaved. If B is any linear Hermitian operator that represents a physically observable property, then the eigenfunctions of gi of B form a complete set.
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If Y(q, t) is the normalized state function of a system at time t, then the average value
of a physical observable B at time t is: <B> = Int (Y* BY dt) The time development of the state of an undisturbed quantum-mechanical system is Given by the Schrodinger time-dependent equation: - (h-bar)/i dY/dt = HY where H is the Hamiltonian (that is, energy) operator of the system.
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