With every physical observable there is associated a mathematical operator which is used in conjunction with the wavefunction. The action of the operator is given by.
The quantum mechanical operator Q associated with a measurable propertu q must be Hermitian. Mathematically this property is defined by.
Physically, the Hermitian property is necessary in order for the measured values eigenvalues to be constrained to real numbers. They can be said to form a basis set in terms of which any wavefunction representing the system can be expressed:.
HERMITIAN OPERATOR QUANTUM MECHANICS I HC VERMA
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. For a mechanical system, the classical Hamiltonian would be just the kinetic energy plus the potential energy, i.
The role of the Hamiltonian in both space and time is contained in the Schrodinger equation. The Operator Postulate With every physical observable there is associated a mathematical operator which is used in conjunction with the wavefunction. 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.
Discussion of operators.
Index Schrodinger equation concepts Postulates of quantum mechanics. Hermitian Property Postulate The quantum mechanical operator Q associated with a measurable propertu q must be Hermitian.
Discussion of expectation values.