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 $\partial_t P +\partial_x j =0$ (*)
I calculated $P=\pi^{-1\over 2}(1+t^2)^{-1\over 2} \exp{({-x^2\over (1+t^2)})}$and $j=ix\pi^{-1\over 2}(1+t^2)^{-3\over 2} \exp{({-x^2\over (1+t^2)})}$
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 do they satisfy (*)? Perhaps I have got something wrong?
Thanks.