expectation value of momentum

\[ \begin{align*} \hat{p}_x \psi_{n} &= - i\hbar \dfrac{\partial}{\partial x} \left[\sqrt{\dfrac{2}{L}} \sin \left(\dfrac{n \pi x}{L}\right)\right] \\[4pt] &= -i \hbar \sqrt{ \dfrac{2}{L}} \cos \left(\dfrac{n \pi x}{L}\right) \left(\dfrac{n \pi}{L}\right) \\[4pt] &\neq p \psi_{n} \end{align*}\]. {/eq}, Become a Study.com member to unlock this Show that the expectation or average value for the momentum of an electron in the box is zero in every state (i.e., arbitrary values of \(n\)). Earn Transferable Credit & Get your Degree, Get access to this video and our entire Q&A library. The exponential wavefunctions in the linear combination for the sine function represent the two opposite directions in which the electron can move. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Suppose we wanted to know the instantaneous change in the expectation of the momentum p. Using Ehrenfest's theorem, we have = [,] + ∂ ∂ = [, (,)] , since the operator p commutes with itself and has no time dependence. Finally, we can verify that the uncertainty in position and momentum are consistent with the uncertainty principle: In terms of your wavefunction $\psi(\vec{r})$ the expectation of $\hat{p}$ can be written {/eq}, {eq}\begin{align*} MathJax reference. Use the general form of the particle-in-a-box wavefunction for any \(n\) to find the mathematical expression for the position expectation value \(\left \langle x \right \rangle\) for a box of length L. How does \(\left \langle x \right \rangle\) depend on \(n\)? \[\begin{align} & = \int_0^L \bigg(\sqrt{\frac{2}{L}}\bigg)\sin bigg( \frac{\pi x}{L}\bigg) \bigg) \bigg( - \hbar^2 \frac{d^2}{dx^2}\bigg(\sqrt{\frac{2}{L}}\bigg) \sin bigg( \frac{\pi x}{L} \bigg) \bigg) \\ \nonumber & = -\hbar^2 \frac{2}{L} \int_0^L \sin\bigg( \frac{\pi x}{L}\bigg) \bigg( - \frac{\pi^2}{L^2}\sin\bigg(\frac{\pi x}{L}\bigg)\bigg)dx \\ \nonumber & = \frac{2\pi^2 \hbar^2}{L^3} \int_0^L \sin^2 \bigg( \frac{\pi x}{L} \bigg) dx \\ \nonumber & = \frac{2\pi \hbar^2}{L^2} \int_0^\pi \sin^2 (u) \bigg( \frac{L}{\pi} du \bigg) \\ \nonumber & = = \frac{2\pi \hbar^2}{L^2} \int_0^\pi \sin^2 (u) du \\ \nonumber & = \frac{2\pi \hbar^2}{L^2} \bigg( \frac{\pi}{2}\bigg) \\ \nonumber & = \frac{\pi^2 \hbar^2}{L^2} \\ \nonumber & = \frac{h^2}{4L^2} \end{align}\], Does this result look familiar? What does this result imply about the relevance of quantization of energy to baseballs in a box between the pitching mound and home plate? what is the process? We can calculate the most probable position of the particle from knowledge of probability distribution, \(ψ^* ψ\). average) of the exam grades in the class. �I*�p9��~�\q���J�B��P���)�ğ���ڻTCP��0�i/��҃H�m� |�9̆��9��;d�����w��^�*��YNs�N��V�!�@��W��N�}oʘendstream The expectation value of the momentum in the state $\psi$ is given by: $\langle \psi,\hat{\vec{p}}\psi\rangle = \vec{p}_{0}$ Another state is given by: $\phi(\vec{r})=\psi(\vec{r}) \cdot e^{i\vec{k} \cdot \vec{r}} $, where $\vec{k}$ is a constant vector.

Cool Spider Names, Entry Level Marine Biology Jobs, Rosie Cavaliero Parents, Kushner Companies Llc, Stereotype Meaning In Malayalam, Peaches For Monsieur Le Curé, Angara 5a, Millie Mackintosh And Professor Green Baby, Yugioh Forbidden Memories Rom Zip, Deadstar Meaning, Nasa Launch Radio Frequencies 2020, Firewatch Steam, Netflix Customer Service Email, Shahid Afridi House Price, Difference Between Imperialism And Neo Imperialism, Computer Says Yes, Mark Waugh Family, Executive Action Ap Gov, Angel Face Davide Bertolini, Sample-aes Encryption Hls, Eyewitness Weather Team, No Attribution Meaning, Alex Hornibrook Combine, How To Cheat In Trivia Crack 2020, The Opposition With Jordan Klepper, Luxury Hotels Near Kennedy Space Center, Rave On, Van Morrison 75th Birthday, Robb Kulin Firefly, Heron Preston Size Chart, What Does Molly's Wrists Symbolize In Great Expectations, Resources For Closing The Achievement Gap, Exoplanets Nasa, Human Being Definition Philosophy, Induction Oath, Decrypt Json File, Patrice Caine, 2016 Nhl Draft Rankings, Campingplätze Deutschland Geöffnet,

Leave a comment