Phy5645/Uncertainty Relations Problem 2
According to the 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 x\,\Delta p \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, we obtain 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 length of a side of the box 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 the size of a nucleon is about 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 10^{-12}\,\text{cm},} that their 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\,\text{MeV}} , and 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 \hbar c\cong 197 \times 10 ^{-13}\,\text{MeV}\cdot\text{cm}} , 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}\,\text{MeV}\cdot\text{cm}) ^2} {(2) (938\,\text{MeV}) (10 ^{-12}\,\text{cm})^2} \approx 0.2\,\text{MeV}}
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