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Heisenberg Uncertainty Principle (view source)
Revision as of 21:13, November 22, 2016
, 21:13, November 22, 2016Added a bit about derivation, will add more
A similar example is [[time]] and [[energy]], the product of which also has a lower limit. Quantum fluctuations are a result of this, where for short time, there is enough energy in "empty space" to create a pair of particle and antiparticle, such as [[electron]] and [[positron]]. Although this appears to be a strange or extreme process, it is the only one to explain some properties of black holes. Laser physics with ultrashort [[laser]] pulses is another field, where the limit in the product of time and energy plays an important role.
A similar example is [[time]] and [[energy]], the product of which also has a lower limit. Quantum fluctuations are a result of this, where for short time, there is enough energy in "empty space" to create a pair of particle and antiparticle, such as [[electron]] and [[positron]]. Although this appears to be a strange or extreme process, it is the only one to explain some properties of black holes. Laser physics with ultrashort [[laser]] pulses is another field, where the limit in the product of time and energy plays an important role.
==Derivation==
It can be shown that for two operators, <math>\hat{A}</math> and <math>\hat{B}</math>, that do not [[Commutative|commute]] that there exists and uncertainty principle:
<math>
\sigma^2_a \sigma^2_b = \frac{1}{4} (\langle {\hat{A} \hat{B}} \rangle - \langle {\hat{B} \hat{A}} \rangle)^2
</math>
For position and momentum, the relevant operators are <math>\hat{x} = x</math> and <math>\hat{p} = \frac{\hbar}{i} \frac{\partial}{\partial x}</math>, and it can be shown that <math>[\hat{x}, \hat{p}] = i \hbar</math>. This leads to the result above.
[[Category:Quantum Mechanics]]
[[Category:Quantum Mechanics]]
[[Category:Chemistry]]
[[Category:Chemistry]]