Table of Contents
HEISENBERG Uncertainty Principle and its Applications
It would be impossible to make accurate measurements of certain pairs of properties, since any technique used for measuring one of them will disturb the system and will cause the measurement of the other one to be imprecise. For instance the method used to measure the position of a particle will result in an uncertainty in the precision of the measurement of the momentum of the particle.
Gedanken experiments are known to be very difficult if not impossible to perform but can easily be carried out in imagination with pretty reliable results. An important gedanken experiment was conducted in 1962 by Werner Heisenberg where he studied ways to make a simultaneous measurement of the position and momentum of a small particle such as an electron. This is what brought about the Uncertainty Principle.
?Px?x ? h / 4? This statement says that in a given direction the product of the uncertainty in the position (?Px) and the uncertainty in the momentum (?x) is approximately equal to the Planck constant (h) divided by 4. The above statement implies that if a particle’s momentum is zero, then the uncertainty of the position of the particle would have to be infinity in order for the product to be greater than or equal to Planck’s constant divided by 4.
Since the momentum p is equal to mu (mass times velocity), the uncertainty expression can also be represented as ?u?x ? h / 4?m.
For all oscillators at temperatures of absolute zero, there exists some residual energy called the zero-point energy. The fact that there is a zero-point energy is consistent with the uncertainty principle; if there was no vibration at absolute zero, the implication would be that both position and momentum of the particles have precise values and this is impossible.
The lifetime-broadening effect deals with widths of spectral lines. The relation ?E?t ? h / 4?m is obtained from the uncertainty principle and variables energy(?E) and time(?t). This statement implies that the energy of a state can only be specified if it has an infinite lifetime. Actually no excited state has an infinite lifetime so no excited state has a precisely defined energy. Because of the uncertainty principle it is impossible to specify exactly the energy levels in a transition. The shorter the lifetime of an excited state, the broader the expected spectral lines. The uncertainty principle is used to estimate the extent of spectral broadening.
The mathematical statement
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Author: djfiske
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