Time independent Klein–Gordon equation with boundary conditions.

[Spinnor]

Say we consider the time independent Klein–Gordon equation, see:

http://en.wikipedia.org/wiki/Klein–Gordon_equation

Lets impose the following boundary conditions, the function is zero at infinity and on some small ball of radius R centered on some origin the function is some complex number C. Assume we have a solution to the time independent Klein–Gordon equation such that psi(R) = C and psi(r=infinity) = 0.

Clearly a global phase change of psi(r) by exp[i*theta],

psi(r) --> psi(r)*exp[i*theta]

does not change the energy in the field psi(r) provided,

C --> C*exp[i*theta]

Is it easy to show that a local phase change which depends only on r,

psi(r) --> psi(r)*exp[i*theta(r)]

will increase the energy of the field? Assume,

C --> C*exp[i*theta(R)]


Thanks for any help!

 

[Spinnor]

Could one just argue that because psi(r) is a solution of the time independent Klein–Gordon equation psi(r) already minimizes energy and any change will only increase the energy?