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
Eigenstates of helicity operator
Reply to thread
Message
[QUOTE="CharlieCW, post: 6051865, member: 649017"] [h2]Homework Statement [/h2] For massless particles, we can take as reference the vector ##p^{\mu}_R=(1,0,0,1)## and note that any vector ##p## can be written as ##p^{\mu}=L(p)^{\mu}_{\nu}p^{\nu}_R##, where ##L(p)## is the Lorentz transform of the form $$L(p)=exp(i\phi J^{(21)})exp(i\theta J^{(13)})exp(i\alpha J^{(30)})$$ Where ##(\theta,\phi)## are the spherical coordinates of ##\vec{p}## and ##\alpha=sinh^{-1}(\frac{1}{2}(p^0-1/p^0))##. This allows to define the general state for the massless particle as: $$|p,\lambda\rangle=U(L(p))|p_R,\lambda\rangle$$ Where ##|p_R,\lambda\rangle## is an eigenstate with value ##\lambda## of the operator ##J_3##. Show that ##|p,\lambda\rangle## is an eigenstate of the helicity operator ##\frac{\vec{p}}{|\vec{p}|}\cdot\vec{J}##. [h2]Homework Equations[/h2] $$J_3|p_R,\lambda\rangle=\lambda|p_R,\lambda\rangle$$ $$\vec{p}=|\vec{p}|(sin\theta cos\phi, sin\theta sin\phi, cos\theta )$$ $$U(\Lambda_a)U(\Lambda_b)=U(\Lambda_a \Lambda_b)$$ [h2]The Attempt at a Solution[/h2] For the last week, I've been trying to verify this last statement by expanding the exponentials or using commutators. For example, by using the commutation relationship $$[J_i,J_k]=i\epsilon_{ijk}J_k$$ But I only end with non-reducible expressions. I also tried expanding the exponentials of the operators using the relationship $$e^{A}=1+A+\frac{1}{2}A^2+\frac{1}{6}A^3+...$$ Without arriving at a result. Particulary, I don't understand how to act using the unitary transformations, as when I even try to start by calculating: $$|p,\lambda\rangle=U(L(p))|p_R,\lambda\rangle)=U(exp(i\phi J^{(21)})exp(i\theta J^{(13)})exp(i\alpha J^{(30)}))|p_R,\lambda\rangle$$ Or even the direct calculation: $$(\frac{\vec{p}}{|\vec{p}|}\cdot\vec{J})|p_R,\lambda\rangle=(\frac{\vec{p}}{|\vec{p}|}\cdot\vec{J})U(L(p))|p_R,\lambda\rangle)$$ I don't know how to reduce terms. Do you have any suggestions? [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Advanced Physics Homework Help
Eigenstates of helicity operator
Back
Top