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(Solved): Consider a scalar field in one spatial dimension described by the Lagrangian L=(1)/(2)\theta _(p)\t ...
Consider a scalar field in one spatial dimension described by the Lagrangian
L=(1)/(2)\theta _(p)\theta ^(\mu )\phi b-V(\phi )=(1)/(2)(\phi ^(˙)^(2)-\phi ^('2))-V(\phi ). (a) If
\phi is a static solution of Lagrange's equation, show that
(d)/(dx)[-(1)/(2)\phi ^('2)+V(\phi )]=0. (b) Suppose
V(\phi )>=0and write
V(\phi )in terms of a function
W(\phi )in the form
V(\phi )=(1)/(2)((d\omega )/(\sigma \phi ))^(2)(explain why this is always possible). If
\phi is a static solution that converges to some solution of the algebraic equation
V(\phi )=0as
x->+-\infty , show that
\phi satisfies the first-order differential equation
(d\phi )/(dx)=+-(dW)/(d\phi )(c) Show that the energy
E=\int_(-\infty )^(\infty ) d\times [(1)/(2)\phi ^(2)+V(\phi )]for a static solution of the type described in part (b) can be written in the form
E=(1)/(2)\int_(-\infty )^(\infty ) dx(\phi ^(')∓(dW)/(d\phi ))^(2)+-[W(\phi (\infty ))-W(\phi (-\infty ))]
=+-[W(\phi (\infty ))-W(\phi (-\infty ))](d) Apply this result to the example discussed in Section 11.6 and get the kink's energy without having to explicitly compute the integral in Eq. (11.136).
