Phy5645/uncertainty relations problem1: Difference between revisions
(New page: This problem taken from Eugen Merzbacher's ''Quantum Mechanics'' 3rd edition: ''Exercise 2.7'' '''Make an estimate of the lower bound for the distance <math>\Delta</math>x, within which a...) |
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For nonrelativistic particles, <math> |t| \ll \frac{m\hbar}{(\Delta p_x)^2} </math>, which can be rearranged to <math> \frac{\Delta p_x}{m}|t| \ll \frac{\hbar}{\Delta p_x} </math>. | |||
Since <math>\frac{\Delta p_x}{m} = \Delta v_x</math>, and <math> \Delta p_x\,\Delta x \cong \hbar \rightarrow \frac{\hbar}{\Delta p_x} \cong \Delta x </math>, we can write: | |||
Since <math>\frac{\Delta p_x}{m} = \Delta v_x</math>, and <math> \Delta p_x \Delta x \cong \hbar \rightarrow \frac{\hbar}{\Delta p_x} \cong \Delta x </math>, we can write: | |||
<math> |t| \Delta \nu_x \cong \Delta x </math>. | <math> |t| \Delta \nu_x \cong \Delta x </math>. | ||
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Replacing <math> \Delta p_x </math> with <math> \Delta p_x = \sqrt{\frac{m\hbar}{|t|}} </math>, the uncertainty in position at time <math>t</math> becomes: | Replacing <math> \Delta p_x </math> with <math> \Delta p_x = \sqrt{\frac{m\hbar}{|t|}} </math>, the uncertainty in position at time <math>t</math> becomes: | ||
<math> \Delta x \approx \sqrt{\frac{\hbar|t|}{m}} = \sqrt{\frac{(1.055\times 10^{-34}J \cdot s)(3.15\times 10^{17}s)}{m}} = \frac{5.77 \times 10^{-9}}{\sqrt{m}}</math>. | <math> \Delta x \approx \sqrt{\frac{\hbar|t|}{m}} = \sqrt{\frac{(1.055\times 10^{-34}\,\text{J}\cdot\text{s})(3.15\times 10^{17}\,\text{s})}{m}} = \frac{5.77 \times 10^{-9}}{\sqrt{m}}</math>. | ||
This is an estimate of the lower bound for the distance within which an object of mass ''m'' can be localized for as long as the universe has existed. | This is an estimate of the lower bound for the distance within which an object of mass ''m'' can be localized for as long as the universe has existed. | ||
We then have the following masses: | We then have the following masses: | ||
<math>m_e = 9.109 \times 10^{-31} kg, m_p = 1.673 \times 10^{-27} kg, 1 g = 0.001 kg, \text{mass of universe} \rightarrow \text{large} </math> | <math>m_e = 9.109 \times 10^{-31}\,\text{kg}, m_p = 1.673 \times 10^{-27}\,\text{kg}, 1 \text{g} = 0.001\,\text{kg}, \text{mass of universe} \rightarrow \text{large} </math> | ||
electron: <math>\frac{5.77 \times 10^{-9}}{\sqrt{9.109 \times 10^{-31}}} = 6.04 \times 10^6\,\text{m}</math> | |||
proton: <math>\frac{5.77 \times 10^{-9}}{\sqrt{1.673 \times 10^{-27}}} = 1.41 \times 10^5\,\text{m}</math> | |||
one-gram object: <math> \frac{5.77 \times 10^{-9}}{\sqrt{0.001}} = 1.82 \times 10^{-7}\,\text{m}</math> | |||
entire universe: <math>\frac{5.77 \times 10^{-9}}{\sqrt{m}}\text{ as m } \rightarrow \text{ large, } \Delta x \rightarrow \text{ very small} </math> | |||
This indicates that an electron and a proton will not be very localized at all. Their initial locations at the start of the universe will indicate very little about their current location. A one-gram object will be much more localized than a proton or electron. An example of a one-gram object is a paper clip. If a paper clip's location is uncertain to <math> 1.82 \times 10^{-7} \text{m},</math> then we would have a very difficult time recognizing this uncertainty in location at all. The mass of the entire universe is not known (since the size of the entire universe is not known either). However, in comparison to a one-gram object, the mass of the universe is very large. As <math>m</math> becomes very large the uncertainty in position becomes very small. The uncertainty in the universe's position becomes negligible. | |||
Back to [[Heisenberg Uncertainty Principle#Problems|Heisenberg Uncertainty Principle]] | |||
Latest revision as of 13:22, 18 January 2014
For nonrelativistic particles, , which can be rearranged to .
Since , and , we can write:
.
Replacing with , the uncertainty in position at time becomes:
.
This is an estimate of the lower bound for the distance within which an object of mass m can be localized for as long as the universe has existed.
We then have the following masses:
electron:
proton:
one-gram object:
entire universe:
This indicates that an electron and a proton will not be very localized at all. Their initial locations at the start of the universe will indicate very little about their current location. A one-gram object will be much more localized than a proton or electron. An example of a one-gram object is a paper clip. If a paper clip's location is uncertain to then we would have a very difficult time recognizing this uncertainty in location at all. The mass of the entire universe is not known (since the size of the entire universe is not known either). However, in comparison to a one-gram object, the mass of the universe is very large. As becomes very large the uncertainty in position becomes very small. The uncertainty in the universe's position becomes negligible.
Back to Heisenberg Uncertainty Principle