Particle Moving on a Straight Line: Where is it Likely to Be Found?

[roshan2004]

A particle moving on a straight line is described by $$\psi(x)=\frac{1+ix}{1+ix^2}$$.
Where is the particle likely to be found?
I took the derivative of probability density with respect to x and equated it to 0. I got my answer to be x=0.643,-0.643,1.554i and -1.554i.
Please tell me whether I am right or wrong or are there any other methods to solve this problem or not?

 

[dx]

There is no reason for the expectation value to be at a stationary point of the probability density.

You have to evaluate the integral <x> = ∫ψ*(x)xψ(x)dx = ∫xP(x)dx.

 

[roshan2004]

But the question is about maximum probability of finding the particle, isn't it?

 

[dx]

"Where is the particle likely to be found" usually means that they want you to find the expectation value of x.

 

[dx]

Unless the exact wording of the question was "where is the particle most likely to be found". Then you would find the x which maximises P(x).

 

[roshan2004]

So if the question is where the particle is most likely to be found, is my answer correct.

 

[dx]

x is a real number, how did you get imaginary values?

 

[roshan2004]

By factorising

 

[dx]

Ok. You have to ignore the imaginary solutions. Looking at this plot:

http://www.wolframalpha.com/input/?i=plot+(1+++x^2)/(1+++x^4)

it does look like the probability density has maxima at approximately x = 0.643 and x = -0.643

 

[roshan2004]

Now I finally got it, thanks dx