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
A simple proof involving degeneracy and commutators
Reply to thread
Message
[QUOTE="loginorsinup, post: 4979259, member: 527065"] [h2]Homework Statement [/h2] In the absence of degeneracy, prove that a sufficient condition for the equation below (1), where [itex]\left|a'\right>[/itex] is an eigenket of [itex]A[/itex], et al., is (2) or (3). [h2]Homework Equations[/h2] [tex]\sum_{b'} \left<c'|b'\right>\left<b'|a'\right>\left<a'|b'\right>\left<b'|c'\right> = \sum_{b',b''} \left<c'|b'\right>\left<b'|a'\right>\left<a'|b''\right>\left<b''|c'\right>\qquad (1)[/tex] [tex]\left[A,B\right] = 0\qquad (2)[/tex] [tex]\left[B,C\right] = 0\qquad (3)[/tex] [h2]The Attempt at a Solution[/h2] I know degeneracy is about one eigenvalue being associated with more than one eigevector (or eigenket). The summation looks intuitive to me. On the right side, a second completeness relation is used, where the completeness relation is [tex]\mathbb{1} = \sum_{b''} \left|b''\right>\left<b''\right|[/tex] I know the commutation relations of (2) and (3) can be written as [tex]\left[X,Y\right] = XY - YX[/tex] for any two operators [itex]X[/itex] and [itex]Y[/itex]. I know that since [itex]\left|a'\right>[/itex] is an eigenket of [itex]A[/itex], [itex]\left|b'\right>[/itex] is an eigenket of [itex]B[/itex] and [tex]\left|c'\right>[/itex] is an eigenket of [itex]C[/itex]. Half the solution is understanding the problem, and the problem seems to be saying that no eigenvalue has the same eigenvector (eigenket). Knowing that prove that the sum is valid if either pair of operators commute. I would greatly appreciate some help moving forward. Thanks! [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Advanced Physics Homework Help
A simple proof involving degeneracy and commutators
Back
Top