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h1. First Postulate -- The State 

The information on a quantum mechanical physical system is represented by a [ket|http://en.wikipedia.org/wiki/Bra-ket_notation] <math>\mid \Psi (t_0) ></math> with the following attributes:
* May be represented by [wavefunction|http://en.wikipedia.org/wiki/Wave_function] <math>\mid \Psi (x,t_0) ></math> or geometrical vector <math>\mid \vec \Psi ></math>
* The states belong to a state vector that has the properties of a vector space
* Properties of the wavefunction <math>\mid \Psi (x,t_0) ></math> are:
** single valued
** [square integrable|http://en.wikipedia.org/wiki/Square_integrable]
** nowhere infinite
** continuous
** piecewise continuous first derivative


Scalar product  (0 if orthogonal):

<math> \left \langle \Psi (x) \mid \phi (x) \right \rangle  \equiv \int_{}^{} \Psi (x)^* \phi (x)\, dx </math>

h1. Second Postulate -- Physical Quantities 

Every measurable quantity "a" is described by an operator A acting on the state space. This operator is [Hermitian|http://www.onethread.org/wiki/index.php?title=Lecture_3#What_are_the_special_properties_of_hermitian_operators.3F] and is called an [observable|http://en.wikipedia.org/wiki/Observable].

h1. Third Postulate -- Real Value Result 

The only possible result of the measurement of a physical quantity "a" is one of the eigenvalues of the corresponding observable A. The measurement of A always gives a real value since A is by definition Hermitian.

h1. Fourth Postulate -- Probability

h2. Discrete Non-Degenerate

When the physical quantity "a" is measured on a system in the normalized state <math>\mid \Psi (t) ></math> , the probability <math>P(a_n)</math> of obtaining the non-degenerate eigenvalue <math>a_n</math> of the corresponding observable is <math>P(a_n)=\mid < \Psi \mid u_n > \mid^2</math>

h2. Continuous Non-Degenerate

When the physical quantity "a" is measured on a system in the normalized state <math>\Psi(x)</math> , the probability dP(a) of obtaining a result between a and a + da is <math>dP(a)=\mid < u_a (x) \mid \Psi (x) > \mid^2 da</math> where <math>u_\alpha (x)</math> is the normalized eigenvector of A associated with the eigenvalue a.

h1. Fifth Postulate -- Measurement Result 

h2. Discrete Non-Degenerate

If the measurement of a physical quantity "a" on a system in the state <math>\Psi(\vec r, t)</math> gives the result <math>a_n</math> . The state of the system immediately after the measurement is <math>u_n (x)</math>.

Consequences:

* The state of the system right after a measurement is always an eigenvector corresponding to the specific eigenvalue that was the result of the measurement.
* The state of the system is fundamentally perturbed by the measurement process.

h1. Sixth Postulate -- Time Evolution

The [time evolution|http://www.onethread.org/wiki/index.php?title=Lecture_3#How_do_we_find_the_time_evolution_of_a_state.3F] of the wavefunction <math>\mid \Psi (x,t)></math> is governed by Schr��dinger's equation <math>\hat H  \Psi = i \hbar \frac{\partial \Psi}{\partial t}
</math>, where H is the Hamiltonian (the observable associated with the total energy of the system).