Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Advanced Physics Homework Help
Quantum Mechanics: Even/Odd Stationary States
Reply to thread
Message
[QUOTE="QuantumBunnii, post: 4318781, member: 406924"] [h2]Homework Statement [/h2] A particle of mass m in the infinite square well of width a at time t = 0 is in a linear superposition of the ground- and the first excited- eigenstates, specifically it has the wave function $$| \Psi(x,t) > = A[ | \psi_1 > + e^{i \phi} | \psi_2 >$$ Find the expectation value <x(t)> of the position of the particle at time t. [h2]Homework Equations[/h2] [tex]< f(x,t) | g(x,t) > = \int_{-a/2}^{a/2} f(x,t)^{*}g(x,t)\ dx[/tex] $$ <x(t)> = < \Psi(x,t) | x \Psi(x,t) > $$ [h2]The Attempt at a Solution[/h2] Obtaining a solution for this problem isn't extraordinarily difficult-- given the stationary states [itex] \psi_1[/itex] and [itex] \psi_2 [/itex], we can simply plug them into the above equations and-- after having normalized for 'A'-- solve for <x(t)>. My question is a more methodological one, regarding the even-ness and odd-ness of the integrands. After going through a few steps to solve for <x>, we would obtain the following solution: $$ <x> = A < \psi_1 | x \psi_1 > + B | \psi_1 | x \psi_2 > + C < \psi_2 | x \psi_1 > + D < \psi_2 | x \psi_2 > $$ where A, B, C, and D, are imaginary constants. In the interest of time, I decided to consider the even-ness and odd-ness of the above integrands: ## \psi_{0,2,4,...} ## correspond to odd functions, and ## \psi_{1,3,5,...} ## correspond to even functions. 'x' should also be considered an odd function, since I have chosen the integral to run from -a/2 to a/2. Hence, we can determine whether each integrand is ultimately even or odd (even times even is even; odd times even is odd; etc.), and immediately set the odd integrands to zero (integrating an odd function centered at the origin yields a value of zero). This would mean: A = 0 and D = 0. However, this turns out to give me the wrong answer, as it seems I'm supposed to consider the value of *all* the integrals. I tried to keep the mathematical calculations in this question to a minimum; I would just like to know whether the above methodology is mathematically intact (I almost hope it isn't...), and-- if so-- why it doesn't work out in this case. Thanks for any help. [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Advanced Physics Homework Help
Quantum Mechanics: Even/Odd Stationary States
Back
Top