Understanding Probability Density for a Bose-Einstein Condensate

[Xgkkp]

Hi,

I'm going through my problem sheet for my quantum mechanics course, and I've come across something I don't know what it means;

It says that a Bose-Einstein condensate can be described by the wave function:

http://xgkkp.no-ip.com/images/math/PsiCondensate.jpg
Psi(x,t) = sqrt( n(x,t) ) * exp ( i*alpha(x,t) )

It then asks me to find the probability density, which I have (assuming it is equal to the (Sigma = |Psi|^2 = Psi.Psi*), and it came out as:

http://xgkkp.no-ip.com/images/math/sigma.jpg
Sigma = n(x,t)

(I can only assume I am correct with this)

The next part has me stumped - it asks me to "Calculate the probability density as a function of n(x,t) and alpha(x,t), and their derivatives".

What does it mean? Does it mean to re-arrange the wave equation so that the functions n(x,t) and a(x,t) are the subject then calculate the probbability density for them? (and the derivative) or does it mean something else entirely?

Thanks,
Nick

ψασ (testing - I don't know if greek letters work with the browser fonts)

 

[slyboy]

Are you sure that both n and alpha are real-valued functions? If alpha is complex then the density would be more complicated.

 

[quantumdude]

Originally posted by Xgkkp
http://xgkkp.no-ip.com/images/math/PsiCondensate.jpg
Psi(x,t) = sqrt( n(x,t) ) * exp ( i*alpha(x,t) )

It then asks me to find the probability density, which I have (assuming it is equal to the (Sigma = |Psi|^2 = Psi.Psi*), and it came out as:

For some reason, I can't view the images you attached, but I have one suggestion.

Take a closer look at the problem and see if it doesn't say to calculate the probability current density, which is:

j=(some factors)[Ψ*(∂/∂x)Ψ-Ψ(∂/∂x)Ψ*]

That's the only way I can see derivatives coming into it.


edit: typo