Question

A three-dimensional wave function of a particle is u(x)=c/r exp(-kr/i)calculate the probability current
density​

Answer 1

Answer:

Calculate the probability current density vector j⃗ for the wave function :

ψ=Ae−i(wt−kx).

From my very poor and beginner's understanding of probability density current it is :

d(ψψ∗)dt=iℏ2m[dψdxψ∗−dψ∗dxψ]

By applying the RHS of the above equation :

iℏ2m[−A2ikxe−i(ωt−kx)ei(ωt−kx)−A2ikxei(ωt−kx)e−i(ωt−kx)]

This gives :

−2iA2ikℏ2m=kℏA2m

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 ?

Answer 2

kℏA2m

Explanation:

Calculate the wave function's probability current density vector j:

ψ=Ae−i(wt−kx).

It's as follows, based on my rudimentary grasp of probability density current:

d(ψψ∗)dt=iℏ2m[dψdxψ∗−dψ∗dxψ]

Using the RHS of the equation above:

iℏ2m[−A2ikxe−i(ωt−kx)ei(ωt−kx)−A2ikxei(ωt−kx)e−i(ωt−kx)]

As a result,

−2iA2ikℏ2m=kℏA2m

A variable number that quantitatively characterises the wave characteristics of a particle. The chance of a particle being present at a given location in space and time is related to the value of its wave function at that location.