The Operator PostulateWith every physical observable there is associated a mathematical operator which is used in conjunction with the wavefunction. Suppose the wavefunction associated with a definite quantized value (eigenvalue) of the observable is denoted by Ψ_{n} (an eigenfunction) and the operator is denoted by Q. The action of the operator is given by The mathematical operator Q extracts the observable value q_{n} by operating upon the wavefunction which represents that particular state of the system. This process has implications about the nature of measurement in a quantum mechanical system. Any wavefunction for the system can be represented as a linear combination of the eigenfunctions Ψ_{n} ( see basis set postulate), so the operator Q can be used to extract a linear combination of eigenvalues multiplied by coefficients related to the probability of their being observed (see expectation value postulate).

Index Schrodinger equation concepts Postulates of quantum mechanics  

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Hermitian Property PostulateThe quantum mechanical operator Q associated with a measurable propertu q must be Hermitian. Mathematically this property is defined by where Ψ_{a} and Ψ_{b} are arbitrary normalizable functions and the integration is over all of space. Physically, the Hermitian property is necessary in order for the measured values (eigenvalues) to be constrained to real numbers. 
Index Schrodinger equation concepts Postulates of quantum mechanics  

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Basis Set PostulateThe set of functions Ψ_{j} which are eigenfunctions of the eigenvalue equation form a complete set of linearly independent functions. They can be said to form a basis set in terms of which any wavefunction representing the system can be expressed: This implies that any wavefunction Ψ representing a physical system can be expressed as a linear combination of the eigenfunctions of any physical observable of the system. 
Index Schrodinger equation concepts Postulates of quantum mechanics  

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Expectation Value PostulateFor a physical system described by a wavefunction Ψ , the expectation value of any physical observable q can be expressed in terms of the corresponding operator Q as follows: It is presumed here that the wavefunction is normalized and that the integration is over all of space. This postulate follows along the lines of the operator postulate and the basis set postulate. The function can be represented as a linear combination of eigenfunctions of Q, and the results of the operation gives the physical values times a probability coefficient. Since the wavefunction is normalized, the integral gives a weighted average of the possible observable values.

Index Schrodinger equation concepts Postulates of quantum mechanics  

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Time Evolution PostulateIf Ψ is the wavefunction for a physical system at an initial time and the system is free of external interactions, then the evolution in time of the wavefunction is given by where H is the Hamiltonian operator formed from the classical Hamiltonian by substituting for the classical observables their corresponding quantum mechanical operators. For a mechanical system, the classical Hamiltonian would be just the kinetic energy plus the potential energy, i.e., the expression for energy. The role of the Hamiltonian in both space and time is contained in the Schrodinger equation. 
Index Schrodinger equation concepts Postulates of quantum mechanics  

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