Phy5645/Uncertainty Relations Problem 2
This a problem from the book of W.Greiner named Quantum Mechanics.
Consider there is a box with a particle (a nucleon) in it. Width of the box is L. Determine the magnitude of kinetic energy of the particle.
According to Heisenberg uncertanity principle: 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 \Delta x \cong \hbar}
and so 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\cong \frac{\hbar}{\Delta x}}
. On the other hand, as we know that 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 E=\frac{\ P^2}{2m}}
. Therefore,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 E= \frac{(\Delta P) ^2}{2m}}
If we plug 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 \ } into the energy equation; it will look 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 E\cong \frac{\hbar ^2}{2m(\Delta x) ^2}}
Let the side of the box shrink 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 L= \Delta x \rightarrow \ 0 }
Knowing that Nucleons have size of 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 ^{12}cm } , Nucleon mass 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 mc^2 \cong 938 MeV} , 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 \hbar c\cong 197 \times 10 ^{-13} cm Mev} , then we can calculate kinetic energy.
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 E\cong \frac{\hbar ^2}{2m(\Delta x) ^2}= \frac{\hbar ^2 c ^2}{2mc ^2(\Delta x) ^2 }= \frac{(197 \times 10 ^{-13} cm Mev) ^2} {(2) (938 MeV) (10 ^{-12}) ^2} \approx 0.2MeV}