Changes

==Principles==

==Principles==

*Every system can be described by a wave function, which is generally a function of the position coordinates and time. All possible predictions of the physical properties of the system can be obtained from the wave function. The wave function can be obtained by solving the [[Schrodinger equation]] for the system.

*Every system can be described by a wave function, which is generally a function of the position coordinates and time. All possible predictions of the physical properties of the system can be obtained from the wave function. The wave function can be obtained by solving the [[Schrodinger equation]] (or the Klein-Gordon equation for relativistic-quantum mechanics) for the system.

*An observable is a property of the system which can be measured. In some systems, many observables can take only certain specific values.  

*An observable is a property of the system which can be measured. In some systems, many observables can take only certain specific values.  

===The uncertainty principle===

===The uncertainty principle===

As a result of the wave nature of a particle, certain quantities cannot be known to an arbitrary precision simultaneously. An example is position and [[momentum (physics)|momentum]]. Whenever its position is measured more accurately (beyond a certain limit), its momentum becomes less certain, and vice versa.  Hence, there is an inherent uncertainty that prevents precisely measuring both the position and the momentum simultaneously. This is known as the [[Heisenberg Uncertainty Principle]]:<ref>http://hyperphysics.phy-astr.gsu.edu/Hbase/uncer.html</ref>

As a result of the wave nature of a particle, certain quantities cannot be known to an arbitrary precision simultaneously. This happens when the  operators for the two quantities do not [[Commutative|commute]]. An example is position and [[momentum (physics)|momentum]]. Whenever its position is measured more accurately (beyond a certain limit), its momentum becomes less certain, and vice versa.  Hence, there is an inherent uncertainty that prevents precisely measuring both the position and the momentum simultaneously. This is known as the [[Heisenberg Uncertainty Principle]]:<ref>http://hyperphysics.phy-astr.gsu.edu/Hbase/uncer.html</ref>

:<math> \sigma_x \sigma_p \ge \frac{\hbar}{2}</math>

:<math> \sigma_x \sigma_p \ge \frac{\hbar}{2}</math>

* <math>\sigma_p</math> is the [[standard deviation]] of momentum

* <math>\sigma_p</math> is the [[standard deviation]] of momentum

* <math>\hbar</math> is the [[Planck's Constant|reduced Planck's constant]]

* <math>\hbar</math> is the [[Planck's Constant|reduced Planck's constant]]

==Interpretations==

==Interpretations==