- #1

da_willem

- 599

- 1

[tex]\tau=t[/tex] and [tex]\sigma =s[/tex]

with s the distance from the origin, and t just time. s runs from -l/2 to l/2. This should be possible right, as I'm free to choose which way to paramterize the string.

Now the solution is (given that the string lies along the x-axis at t=0) ofcourse:

[tex] \vec{X} (t,s) = s(cos(\omega t), sin(\omega t))[/tex]

As the endpoints are not tied down to anything I have to use free enpoint conditions which means the endpoints move at the speed of light. This yields for the enpoints [tex]c= \omega l/2[/tex]. Futhermore you can calculate that l is related to the energy of the string E by:

[tex]\frac{2}{\pi} \frac{E}{T_0}[/tex]

With T0 the tension of the string. Now the solution is complete as both l and omega are found.

Now I would like to calculate the angular momentum, so I can find [itex]\alpha '[/tex] the constant of proprtianality between the angular momentum and the energy squared (in terms of hbar). I tried:

[tex]J= \int _{-l/2} ^{l/2} X_1 P_2 - X_2 P_1 ds [/tex] should be [tex]\frac{E^2}{2 \pi T_0 x}[/tex]

but now I don't know what expression to use for P the momentum density. I tried:

[tex]\vec{P} =\frac{T_0}{c^2} \frac{\partial \vec{X}}{\partial t}[/tex]

but this yields a result a factor 4/3pi off. So I guess there is something wrong with my expression for P. [itex]T_0/c^2[/itex] is the rest mass density of a relativistic string. But my string is moving, so I tried

[tex]\vec{P}=\gamma (v) \frac{T_0}{c^2} \frac{\partial \vec{X}}{\partial t}[/tex]

with [itex]v= \omega s[/itex], but this diverges bacause gamma explodes at the string endpoints. What's wrong? How to calculate the angular momentum using [itex] \sigma =s[/itex]?