Phy5645/uncertainty relations problem1
This problem taken from Eugen Merzbacher's Quantum Mechanics 3rd edition: Exercise 2.7
Make an estimate of the lower bound for the distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta} x, within which an object of mass m can be localized for as long as the universe has existed (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \approx 10^{10} } years). Compute and compare the values of this bound for an electron, a proton, a one-gram object, and the entire universe.
For nonrelativistic particles: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |t| \ll \frac{m\hbar}{(\Delta p_x)^2} } , which can be rearranged to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\Delta p_x}{m}|t| \ll \frac{\hbar}{\Delta p_x} } .
Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\Delta p_x}{m} = \Delta v_x} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta p_x \Delta x \cong \hbar \rightarrow \frac{\hbar}{\Delta p_x} \cong \Delta x } , we can write:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |t| \Delta \nu_x \cong \Delta x } .
Replacing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta p_x } with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta p_x = \sqrt{\frac{m\hbar}{|t|}} } , the uncertainty in position at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} becomes:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}} .
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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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} }
electron: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{5.77 \times 10^{-9}}{\sqrt{9.109 \times 10^{-31}}} = 6.04 \times 10^6 \text{m}}
proton: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{5.77 \times 10^{-9}}{\sqrt{1.673 \times 10^{-27}}} = 1.41 \times 10^5 \text{m}}
one-gram object: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{5.77 \times 10^{-9}}{\sqrt{0.001}} = 1.82 \times 10^{-7} \text{m}}
entire universe: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{5.77 \times 10^{-9}}{\sqrt{m}}\text{ as m } \rightarrow \text{ large, } \Delta x \rightarrow \text{ very small} }
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1.82 \times 10^{-7} \text{m}} , 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 very large. As m becomes very large the uncertainty in position becomes very very small. The uncertainty in the universe's position becomes negligible.