Calculate the probability current density vector $\vec{j}$ for the wave function : $$\psi = Ae^{-i(wt-kx)}.$$
From my very poor and beginner's understanding of probability density current it is :
$$\frac{d(\psi \psi^{*})}{dt}=\frac{i\hbar}{2m}[\frac{d\psi}{dx}\psi^{*}-\frac{d\psi^{*}}{dx}\psi]$$
By applying the RHS of the above equation :
$$\frac{i\hbar}{2m}[-A^{2}ikxe^{-i(ωt-kx)}e^{i(ωt-kx)}-A^{2}ikxe^{i(ωt-kx)}e^{-i(ωt-kx)}]$$
This gives :
$$\frac{-2iA^{2}ik\hbar}{2m}=\frac{k \hbar A^{2}}{m}$$
This is not the correct answer. :( What have I done wrong ?
In the model workings instead of A in the complex conjugate of the wave function they have written $A^{*}$. Why is this necessary since $A$ is likely to be a real number anyways ?
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