The Basic Postulate of Quantum Mechanics

Now we can determine states for which <DA²> = 0; A is the exact value (precise measurement, for instantaneous momentum)

The wavefunction that satisfies an equation py = py  or the wavefunction which satisfies an equation Hy = E y is called the eigenfunction and A (p and E here) is the eigenvalue.
 

for instance y = e^ax; A = ^d/_dx  →^dy/_dx = a              a: Eigenvalue     y: Eigenfunction

The Basic Postulate of Quantum Mechanics:
 

In general the equation Ay = Ay has its solutions only for definite values of physical quantity A. They produce either discrete series A₁, A₂,.... or continuous series of values in the corresponding range. The measurement of a physical value A (for instance, measurement of energy E) in a state y_A1 gives the exact value of A₁ (for instance, the definite value of energy E₁).

The precisely accurate physical property, or eigen value, of a system can be produced mathematically when the system is characterized by an eigenfunction. If we repeat our measurements of this value we will continue to calculate the same eigenvalue. The particle under consideration is in defined state and its eigenfunction is acted on by the corresponding operator.

The total energy E average of distribution is in general:
 

<E>  =  -¥ò+∞y*	(−h²/2mΔ +	V(x)) y dx
	ê	|
kinetic energy	¿     +	èpotential energy

If we compare all such equations we will be able to produce a rule for calculating the quantity A, the average of distribution in quantum mechanics:
 

A is the operator here, which corresponds to physical quantity A. In the Dirac's style one can obtain:
 

If we consider the normal energy average of distribution: <E> = _-¥ò^+∞y*Hydx ; where y is the Schroedinger equation solution Hy = Ey, then it's obviously:   <E>  =  _-¥ò^+∞y*Hydx = E · òy* ydx = E. I.e. the average of distribution is apparently the energy value E! There is no overlapping and hence no uncertainty. This is so because E is the eigenvalue of the energy operator H. 
 

Example:

We when measure the momentum of a particle we always obtain the measured value p. This means that now the particle is in the state in which the eigenfunction is the momentum operator. The wavefunction after making momentum measurements is as follows

since the eigenvalue equation for momentum py_p= p y_p:
 

If we now measure the kinetic energy then this measurement corresponds mathematically to the application of the kinetic energy operator, −^h²/_2m^∂²/_∂x², on the wavefunction y_p:

It finally gives us the eigenvalue  ^p²/_2m. This means that the wave keeps like wavefunction y_p. The kinetic energy measurement doesn't annihilate the result of first measurement. Hence it is possible to make two measurements simultaneously with any degree accuracy. 

If y isn't an eigenfunction then it's possible to expand y on eigenfunctions φ_n with coefficients a_n in the following way:

Ok, but what are the transformation coefficients a_n?

Multiplication by <m| to the left gives:
 

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