- #1

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## Homework Statement

consider a particle at an interval ##[-L/2, L/2]##, described by the wave function ## \psi (x,t)= \frac{1}{\sqrt{L}}e^{i(kx-wt)}##

a) Calculate the probability density ##\rho (x,t) ## and the current density ## j(x,t)## of the particle

b) How can you express ## j(x,t)## as a function of ##\rho (x,t) ## and the velocity ## v##?

c) What is de probability to find the particle at the intervals ##[-L/2, L/2]##, ##[-L/2, 0]## and ##[0, L/4]##

(note: not sure whether 'current density' is the proper translation of the term I have in Dutch for j(x,t).

## Homework Equations

##\rho (x,t) = \psi^* (x,t) \psi (x,t)##

## j(x,t) = - \frac{i \hbar}{2m} \big[ \psi^* (x,t) \frac{\partial}{\partial x}\psi (x,t) - \psi (x,t) \frac{\partial}{\partial x}\psi^* (x,t) \big]##

##P(\Omega)= \int_{\Omega} |\psi (x,t)|^2 dx##

no clue on the relevant equation for b and c

## The Attempt at a Solution

a) ##\rho (x,t) = \frac{1}{\sqrt{L}} e^{-i(kx-wt)} . \frac{1}{\sqrt{L}}e^{i(kx-wt)} = \frac{1}{L} e^{-i(kx-wt)+i(kx-wt)} = \frac{1}{L} e^0 = \frac{1}{L} ##

## j(x,t) = - \frac{i \hbar}{2m} \big[ \frac{1}{\sqrt{L}} e^{-i(kx-wt)} \frac{\partial}{\partial x}\frac{1}{\sqrt{L}}e^{i(kx-wt)} - \frac{1}{\sqrt{L}} e^{i(kx-wt)} \frac{\partial}{\partial x}\frac{1}{\sqrt{L}}e^{-i(kx-wt)} \big]##

## = - \frac{i \hbar}{2m} \big[ \frac{1}{L} e^{-i(kx-wt)} e^{i(kx-wt)} ik - \frac{1}{L} e^{-i(kx-wt)} e^{i(kx-wt)} (-ik) \big]##

## = - \frac{i \hbar}{2m} \big[ \frac{ik}{L} + \frac{ik}{L} \big]##

## = - \frac{i \hbar}{m} \frac{ik}{L} = \frac{\hbar k}{m L}##

b) I was not sure how to interprete the question, express first ## j(x,t)## as a function of ##\rho (x,t) ## AND afterwards in relation to velocity, OR all at the same time?

I have suggested for myself at least that ## j(x,t) = \frac{\hbar k}{m} \rho (x,t) ##

I have no idea how the velocity comes into that.

c) For the last part I have

## \frac{1}{L} \int_{-L/2}^{L/2} dx = \frac{x}{L}|_{-L/2}^{L/2} = 2L/2L =1##

Since the second interval is half of the first interval the probability should be 1/2, and checking the integral that is what I got. Same logic applies for the third given interval, which is a 4th of the first, and the probability is 1/4.

Can someone please review what I did solve, and whether I applied the correct logic or proper techniques. And can someone help me out with the velocity part?