POSTULATES [1]

Postulate I.
To any self-consistently and well-observable in physics (A), such as linear momentum, energy, mass, angular momentum, or number of particles, there corresponds an operator such that measurement of A yields values (a) which are eigenvalues of . The corresponding eigenvalue equation

The function is called the eigenfunction of corresponding to the eigenvalue a.

Postulate II.
Measurement of the observable A that yields the value a leaves the system in the state , where is the eigenfunction of that corresponds to the eigenvalue a.

Postulate III.
The state of a system at any instant of time may be represented by a state or wave function which is continuous and differentiable. All information regarding the state of the system is cocntained in the wavefunction. The average, , which is called the expectation value of C is

The observable C is measured in a specific experiment, X. There are a large number (N) of identical replicas of X. The initial states in each such replica are all identical. At the time t, one measures C in all these replica experiments and obtains the set of values Cs. The average of C is then given by

where N >> 1

Postulate IV.
The state function for a system (i.e., a single particle system) develops in time according to the equation:

where H is the Hamiltonian. This equation is called the time-dependent Schrodinger equation.

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POSTULATES [2]

Postulate I.
To an ensemble of physical systems one can, in certain cases, associate a wave function or state function which contains all the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an arbitrary complex number without altering its physical significance.

Postulate II.
The superposition principle. The dynamical states of a quantum system are linearly superposable.

where is associated with one possible state of a statistical ensemble of physical systems and c is complex constants of state function associated with a possible state of the ensemble.

Postulate III.
With every dynamical variable is associated a linear operator.

Postulate IV.
The only result of a precise measurement of the dynamical variable A is one of the the eigenvalues of the linear operator associated with A.

Postulate V.
If a series of measurements is made of the dynamical variable A on an ensemble of system, described by the wave function , the expectation or average value of this dynamical variable is

the average value is real if C is a Hermitian. If the wave function is normalized to unity, then we have

Postulate VI.
A wave function representing any dynamical state can be expressed as a linear combination of the eigenfunctions of .

Postulate VII.
The time evolution of the wave function of a system is determined by the time-dependent Schrodinger equation

where is the Hamiltonian, or total energy operator of the systems.

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POSTULATES [3]

Postulate I.
For any possible state of a system, there is a function, , of the coordinates of the parts of the system and time that completely describes the system.

Postulate II.
For every dynamical variable (classical observable) there is a corresponding operator.

Postulate III.
The permissible values that a dynamical variable may have are those given by , where is the eigenfunction of the operator that corresponds to the observable whoise permissible values are a.

Postulate IV.
The state function, , is given as a solution of

where is the operator of the total energy the Hamiltonian operator.

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Some important nonlinear differential equations that are solved by series techniques

Hermite polynomials

Bessel functions

Legendre polynomials

Chebyshev polynomials

Laguerre polynomials

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Reference:
1. Richard L. Liboff, Introductory Quantum Mechanics (3rd Edition, 1997), Addison Wesley Longman.
2. B.H. Bransden and C. J. Joachain, Quantum Mechanics (1999), Prentice Hall.
3. J. E. House, Fundamentals of Quantum Mechanics (1998), Academic Press.
4. J. J. Sakurai, Modern Quantum Mechanics (1985), Addison Wesley.