1. The problem statement, all variables and given/known data
2. Relevant equations
\begin{align}
\Delta p \Delta x &\geq \frac{\hbar}{2}\\
\langle p \rangle &= \sqrt{\langle p^2 \rangle - \langle p \rangle^2}\\
\langle x \rangle &= \sqrt{\langle x^2 \rangle - \langle x \rangle^2}
\end{align}
I allso know that a wavefunction of a particle in an infinite square well is: ##\psi=\sqrt{\frac{2}{d}}\sin\left(\frac{N\pi}{d}x\right)##, but i don't know ##N##. From this it can be derived that ##\langle p\rangle = 0## and ##\langle p^2\rangle = \frac{\hbar^2\pi^2}{d^2}N^2=E2m##. Too bad i don't know ##N## again...
3. The attempt at a solution
I first assumed that uncertainty in position ##\Delta x=d## and then calculated a momentum ##\Delta p=\frac{\hbar}{2\Delta x}=9.845MeV/c##?
If I try to calculate uncertainty in full energy relativisticaly I start with a Lorentz invariance:
\begin{align}
E^2&=p^2c^2+{E_0}^2\\
E&=\sqrt{p^2c^2+{E_0}^2}\\
&\left\downarrow\substack{\text{Because energy $E$ is a function}\\\text{of only one variable $p$, we use}\\\text{standard formula for calculating}\\\text{uncertainty}.}\right. \quad \boxed{\Delta q = \frac{dq}{dp}\Delta p}\\
\Delta E &= \frac{d}{dp}\left(\sqrt{p^2c^2+{E_0}^2}\right)\cdot\Delta p\\
\Delta E &= \frac{1}{2}\frac{1}{\sqrt{p^2c^2+{E_0}^2}}2c^2p\cdot \Delta p\\
\Delta E &= \frac{c^2p}{\sqrt{p^2c^2+{E_0}^2}}\cdot \Delta p\\
\end{align}
I did all by the book but I can't seem to calculate ##\Delta E## as I have to get rid of ##p## (which is unknowne) somehow. I get even more lost when trying to calculate the uncertainty ##\Delta E_k##:
\begin{align}
E^2 &= p^2c^2 + {E_0}^2\\
E &= \sqrt{p^2c^2 + {E_0}^2}\\
E_k + E_0 &= \sqrt{p^2c^2 + {E_0}^2}\\
E_k &= \sqrt{p^2c^2 + {E_0}^2} - E_0\\
&\downarrow\\
\Delta E_k &= \frac{d}{dp}\left(\sqrt{p^2c^2 + {E_0}^2} - E_0\right) \cdot \Delta p\\
\Delta E_k &= \frac{d}{dp}\left(\sqrt{p^2c^2 + {E_0}^2} - E_0\right) \cdot \Delta p\\
\Delta E_k &= \frac{1}{2}\frac{1}{\sqrt{p^2c^2+{E_0}^2}}2c^2p\cdot \Delta p\\
\Delta E_k &= \frac{c^2p}{\sqrt{p^2c^2+{E_0}^2}}\cdot \Delta p\\
\end{align}
From neither of the equations I could calculate ##\Delta E## or ##\Delta E_k## but I have noticed that they are the same. Which makes sense.
What could I do to be able to calculate ##\Delta E## or ##\Delta E_k##?
Quote:
Calculate uncertainty in energy ##\Delta E## or kinetic energy ##\Delta E_k## if we only know that an electron is closed in a 1-D box of width ##d=10fm##. |
2. Relevant equations
\begin{align}
\Delta p \Delta x &\geq \frac{\hbar}{2}\\
\langle p \rangle &= \sqrt{\langle p^2 \rangle - \langle p \rangle^2}\\
\langle x \rangle &= \sqrt{\langle x^2 \rangle - \langle x \rangle^2}
\end{align}
I allso know that a wavefunction of a particle in an infinite square well is: ##\psi=\sqrt{\frac{2}{d}}\sin\left(\frac{N\pi}{d}x\right)##, but i don't know ##N##. From this it can be derived that ##\langle p\rangle = 0## and ##\langle p^2\rangle = \frac{\hbar^2\pi^2}{d^2}N^2=E2m##. Too bad i don't know ##N## again...
3. The attempt at a solution
I first assumed that uncertainty in position ##\Delta x=d## and then calculated a momentum ##\Delta p=\frac{\hbar}{2\Delta x}=9.845MeV/c##?
If I try to calculate uncertainty in full energy relativisticaly I start with a Lorentz invariance:
\begin{align}
E^2&=p^2c^2+{E_0}^2\\
E&=\sqrt{p^2c^2+{E_0}^2}\\
&\left\downarrow\substack{\text{Because energy $E$ is a function}\\\text{of only one variable $p$, we use}\\\text{standard formula for calculating}\\\text{uncertainty}.}\right. \quad \boxed{\Delta q = \frac{dq}{dp}\Delta p}\\
\Delta E &= \frac{d}{dp}\left(\sqrt{p^2c^2+{E_0}^2}\right)\cdot\Delta p\\
\Delta E &= \frac{1}{2}\frac{1}{\sqrt{p^2c^2+{E_0}^2}}2c^2p\cdot \Delta p\\
\Delta E &= \frac{c^2p}{\sqrt{p^2c^2+{E_0}^2}}\cdot \Delta p\\
\end{align}
I did all by the book but I can't seem to calculate ##\Delta E## as I have to get rid of ##p## (which is unknowne) somehow. I get even more lost when trying to calculate the uncertainty ##\Delta E_k##:
\begin{align}
E^2 &= p^2c^2 + {E_0}^2\\
E &= \sqrt{p^2c^2 + {E_0}^2}\\
E_k + E_0 &= \sqrt{p^2c^2 + {E_0}^2}\\
E_k &= \sqrt{p^2c^2 + {E_0}^2} - E_0\\
&\downarrow\\
\Delta E_k &= \frac{d}{dp}\left(\sqrt{p^2c^2 + {E_0}^2} - E_0\right) \cdot \Delta p\\
\Delta E_k &= \frac{d}{dp}\left(\sqrt{p^2c^2 + {E_0}^2} - E_0\right) \cdot \Delta p\\
\Delta E_k &= \frac{1}{2}\frac{1}{\sqrt{p^2c^2+{E_0}^2}}2c^2p\cdot \Delta p\\
\Delta E_k &= \frac{c^2p}{\sqrt{p^2c^2+{E_0}^2}}\cdot \Delta p\\
\end{align}
From neither of the equations I could calculate ##\Delta E## or ##\Delta E_k## but I have noticed that they are the same. Which makes sense.
What could I do to be able to calculate ##\Delta E## or ##\Delta E_k##?
via Physics Forums RSS Feed http://www.physicsforums.com/showthread.php?t=703329&goto=newpost
0 commentaires:
Enregistrer un commentaire