Does the given wavefunction satisfy conservation of probability?

[c299792458]

Homework Statement

I have a wavefunction \psi = \pi^{-1\over 4}(1+it)^{-1\over 2} \exp{({-x^2\over 2(1+it)})}

I want to show that it satisfies the conservation of probability.

Homework Equations

\partial_t P +\partial_x j =0 --(*)

The Attempt at a Solution

I calculated the probability distribution to be P=\pi^{-1\over 2}(1+t^2)^{-1\over 2} \exp{({-x^2\over (1+t^2)})} and the probability current j=ix\pi^{-1\over 2}(1+t^2)^{-3\over 2} \exp{({-x^2\over (1+t^2)})}

This gives \partial_t P = -t\pi^{-1\over 2}(1+t^2)^{-5\over 2} (t^2-2x^2+1)\exp{({-x^2\over (1+t^2)})} and \partial_x j = i\pi^{-1\over 2}(1+t^2)^{-5\over 2} (t^2-2x^2+1)\exp{({-x^2\over (1+t^2)})}

But how are they do they satisfy (*)?

tHanks!

 

[grzz]

\partial_t P +\partial_x j =0 --(*)


This gives \partial_t P = -t\pi^{-1\over 2}(1+t^2)^{-5\over 2} (t^2-2x^2+1)\exp{({-x^2\over (1+t^2)})} and \partial_x j = i\pi^{-1\over 2}(1+t^2)^{-5\over 2} (t^2-2x^2+1)\exp{({-x^2\over (1+t^2)})}

But how are they do they satisfy (*)?

If equation (*) is satisfied then probability IS conserved. And your derivatives of P and of j show just that.

 

[c299792458]

@grzz:

Thanks, I know that! The problem is notice how the 2 partial derivatives are not exactly equal! My aim is to fit them into (*).

 

[grzz]

Sorry! I missed the extra t in the derivative of P.

 

[c299792458]

@grzz:

Also the extra i in the 2nd expression :(