Colby College Postulates of Quantum Mechanics 1 I. The physical state of the system is described by a wave function as completely as possible. The wave function is derived from an orthonormal set of eigenfunctions of the Hamiltonian. Example : Particle-in-a-Box: Ψ n (x) = (2/a) ½ sin(n π x/a) are the solutions to H ^ Ψ=ΕΨ II. Any observable may be represented by a linear operator. The results should be a real number. The least restrictive requirement is that the operator must be Hermitian: ∫ Ψ * j o ^ Ψ i dx = ∫ Ψ i (o ^ Ψ j ) * dx = ∫ Ψ i o ^* Ψ * j dx The observable operator is constructed from the following table: Classical Quantum Operator x x ^ p h – i ∂ ∂ x = – ih- ∂ ∂ x t t ^ E vs. time E ^ = ih- ∂ ∂ t III. If the wave function is an eigenfunction of the observable, then repeated measurements of the observable always give the same result, the eigenvalue: if o ^ Ψ = o Ψ then each measurement gives the result, o. Example
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