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 :(