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8: The Postulates of Quantum Mechanics

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    • 8.1: The Classical Wave Equation
      The mathematical description of the one-dimensional waves can be expressed as solutions to the "wave equation." It may not be surprising that not all possible waves will satisfy the wave equation for a specific system since waves solutions must satisfy both the initial conditions and the boundary conditions. This results in a subset of possible solutions. In the quantum world, this means that the boundary conditions are responsible somehow for the quantization phenomena in Chapter 1.
    • 8.2: The Schrödinger Equation
      Erwin Schrödinger posited an equation that predicts both the allowed energies of a system as well as address the wave-particle duality of matter. Schrödinger equation for de Broglie's matter waves cannot be derived from some other principle since it constitutes a fundamental law of nature. Its correctness can be judged only by its subsequent agreement with observed phenomena (a posteriori proof).
    • 8.3: Operators in Quantum Mechanics
      An operator is a generalization of the concept of a function. Whereas a function is a rule for turning one number into another, an operator is a rule for turning one function into another function.
    • 8.4: Quantum Math
      To every dynamical variable in quantum mechanics, there corresponds an eigenvalue equation . The eigenvalues represents the possible measured values of the operator.
    • 8.5: Postulate 1 of Quantum Mechanics
      Postulate 1: Every physically-realizable state of the system is described in quantum mechanics by a state function that contains all accessible physical information about the system in that state.
    • 8.6: Postulate 2 of Quantum Mechanics
      Every observable in quantum mechanics is represented by an operator which is used to obtain physical information about the observable from the state function. For an observable that is represented in classical physics by a function \(Q(x,p)\), the corresponding operator is \(Q(\hat{x},\hat{p})\).
    • 8.7: Postulates 3 and 4 of Quantum Mechanics
      It is a general principle of Quantum Mechanics that there is an operator for every physical observable. A physical observable is anything that can be measured. If the wavefunction that describes a system is an eigenfunction of an operator, then the value of the associated observable is extracted from the eigenfunction by operating on the eigenfunction with the appropriate operator. The value of the observable for the system is the eigenvalue, and the system is said to be in an eigenstate.
    • 8.8: Postulate 5 of Quantum Mechanics
      While the time-dependent Schrödinger equation predicts that wavefunctions can form standing waves (i.e., stationary states), that if classified and understood, becomes easier to solve the time-dependent Schrödinger equation for any state. Stationary states can also be described by the time-independent Schrödinger equation (used when the Hamiltonian is not explicitly time dependent). The solutions to the time-independent Schrödinger equation still have a time dependency.
    • 8.9: Postulates and Principles of Quantum Mechanics (Exercises)
      These are homework exercises to accompany Chapter 4 of McQuarrie and Simon's "Physical Chemistry: A Molecular Approach" Textmap.

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