First Postulate – The State

The information on a quantum mechanical physical system is represented by a ket <math>\mid \Psi (t_0) ></math> with the following attributes:

• May be represented by wavefunction <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
  • nowhere infinite
  • continuous
  • piecewise continuous first derivative

Scalar product (0 if orthogonal):

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

Second Postulate – Physical Quantities

Every measurable quantity "a" is described by an operator A acting on the state space. This operator is Hermitian and is called an observable.

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.

Fourth Postulate – Probability

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>

Continuous Non-Degenerate

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

Fifth Postulate – Measurement Result

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 </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.

Sixth Postulate – Time Evolution

The time evolution of the wavefunction <math>\mid \Psi (x,t)></math> is governed by Schr��dinger's equation <math>\hat H \Psi = i \hbar \frac

</math>, where H is the Hamiltonian (the observable associated with the total energy of the system).