- #1
opous
- 16
- 0
[SOLVED] Klein Gordon equation, probability density
Use the Klein-Gordon Equation to show that
\partial_{\mu}j^{\mu} = 0
KG:
\left(\frac{\partial^{2}}{\partial t^{2}} - \nabla^{2} + m^{2}\right) \phi = (\partial_{\mu}\partial^{\mu} + m^{2}) \phi = 0
j:
j^{\mu} = \frac{i}{2m} \left[\phi^{*}(\partial^{\mu} \phi) - (\partial^{\mu} \phi^{*}) \phi]\right
OK, so I've expanded j out, taking the differentials and have the following for my components:
time component:
\frac{i}{2m} \left[\phi^{*} \frac{\partial^{2} \phi}{\partial t^{2}} - \phi \frac{\partial^{2} \phi^{*}}{\partial t^{2}}\right]
space components:
\frac{i}{2m} \left[- \phi^{*} \frac{\partial^{2} \phi}{\partial r^{2}} + \phi \frac{\partial^{2} \phi^{*}}{\partial r^{2}}\right]
(where I've used r = (x,y,z) )
Putting them in this form:
\frac{i \phi^{*}}{2m} \left[\frac{\partial^{2} \phi}{\partial t^{2}} - \frac{\partial^{2} \phi}{\partial r^{2}}\right] - \frac{i \phi}{2m} \left[\frac{\partial^{2} \phi^{*}}{\partial t^{2}} - \frac{\partial^{2} \phi^{*}}{\partial r^{2}}\right]
Which looks slightly more like the KG equation. However, my trouble is with the mass terms, I have no (mass)² terms to complete the KG equation and hence show the desired expression is zero. Can anyone see what I'm missing here?
Homework Statement
Use the Klein-Gordon Equation to show that
\partial_{\mu}j^{\mu} = 0
Homework Equations
KG:
\left(\frac{\partial^{2}}{\partial t^{2}} - \nabla^{2} + m^{2}\right) \phi = (\partial_{\mu}\partial^{\mu} + m^{2}) \phi = 0
j:
j^{\mu} = \frac{i}{2m} \left[\phi^{*}(\partial^{\mu} \phi) - (\partial^{\mu} \phi^{*}) \phi]\right
The Attempt at a Solution
OK, so I've expanded j out, taking the differentials and have the following for my components:
time component:
\frac{i}{2m} \left[\phi^{*} \frac{\partial^{2} \phi}{\partial t^{2}} - \phi \frac{\partial^{2} \phi^{*}}{\partial t^{2}}\right]
space components:
\frac{i}{2m} \left[- \phi^{*} \frac{\partial^{2} \phi}{\partial r^{2}} + \phi \frac{\partial^{2} \phi^{*}}{\partial r^{2}}\right]
(where I've used r = (x,y,z) )
Putting them in this form:
\frac{i \phi^{*}}{2m} \left[\frac{\partial^{2} \phi}{\partial t^{2}} - \frac{\partial^{2} \phi}{\partial r^{2}}\right] - \frac{i \phi}{2m} \left[\frac{\partial^{2} \phi^{*}}{\partial t^{2}} - \frac{\partial^{2} \phi^{*}}{\partial r^{2}}\right]
Which looks slightly more like the KG equation. However, my trouble is with the mass terms, I have no (mass)² terms to complete the KG equation and hence show the desired expression is zero. Can anyone see what I'm missing here?