Klein-Gordon Equation & Continuity Equation

[Sekonda]

Hello,

My question is on the Klein-Gordon equation and it's relation to the continuity equation, so for a Klein-Gordon equation & continuity equation of the following form, I have attained the following probability density and probability current relations (although not normalised correctly? If that's the correct term):

$$-\frac{\partial^2 }{\partial t^2}\psi +\bigtriangledown ^2\psi-m^2\psi=0 \: ,\: \frac{\partial }{\partial t}\rho +\mathbf{\bigtriangledown } \cdot \mathbf{j}=0$$

$$\rho=i(\psi^{*}\dot{\psi}-\psi\dot{\psi^{*}})\,,\; \mathbf{j} =i(\psi\bigtriangledown \psi^{*}-\psi^{*}\bigtriangledown\psi)$$

We then impose conservation of probability by making the time derivative of the probability equal to zero, and the following manipulation of this derivative is shown below:

$$\frac{\partial P}{\partial t}=\int_{V}\frac{\partial \rho}{\partial t}\, d^3x=-\int_{V}\bigtriangledown\cdot \mathbf{j}\, d^3x=-\oint_{S}\mathbf{j}\cdot d\mathbf{s}=0$$

This above equation makes use of the divergence theorem in the last pair. This is true if j=0, but does j in general need to be equal to zero? I thought 'j' would vary around the surface and so the integration of all 'j' at each part on the surface would equal zero but not necessarily all 'j' =0.

To be honest I'm quite unsure at what 'j' represents in terms of probability!

Can anyone explain this?

Thanks,
SK

 

[Sekonda]

Also I want to show that the probability density is not always positive but can be negative, so I tried substituting a plane wave solution of the form:

$$\psi\sim e^{-iEt+i\mathbf{p\cdot x}}$$

into my probability density equation in the post above, but this gave me a positive answer - though if I use the negative energy plane wave solutions I will attain negative probability density (negative energy plane wave solutions attained by switching the signs in the exponent above), is this the correct way of showing this?

The -ve energies correspond to the -ve probabilities?

 

[Demystifier]

Sekonda, if you take a SUPERPOSITION of two plane waves with different positive energies, you will obtain that rho is not positive at some points. This implies that rho is NOT a probability density.

For some details, see also
http://arxiv.org/abs/0804.4564

 

[andrien]

there is a problem with single particle representation of klein-gordon eqn.you can see that ρ depends on both ψ and ∂ψ/∂t,and klein-gordon eqn itself is a second order differential eqn which requires both ψ and ∂ψ/∂t to be specified at say t=0.so they can take arbitrary value and hence can make ρ negative.it is here identified as charge density rather than probability density.

 

[Sekonda]

Thanks guys,

I think that link is a bit too complex for me but I think I understand how to go about the superposition of waves, I'll try it. Though is my negative energy plane-wave not also a simple and fair enough way of showing that the probability density can be negative - but this only occurs for negative energy solutions.

Cheers,
SK