Understanding Probability Density for a Bose-Einstein Condensate

In summary: Thanks for catching that! In summary, a Bose-Einstein condensate can be described by the wave function: -It has a Sigma (|Psi|^2) value-The probability density is calculated as a function of n(x,t) and alpha(x,t), and their derivatives.
  • #1
Xgkkp
3
0
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 [Broken]
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 [Broken]
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)
 
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  • #2
Are you sure that both n and alpha are real-valued functions? If alpha is complex then the density would be more complicated.
 
  • #3
Originally posted by Xgkkp
http://xgkkp.no-ip.com/images/math/PsiCondensate.jpg [Broken]
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
 
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1. What is Qm?

Qm, or quantum mechanics, is a branch of physics that studies the behavior and interactions of matter and energy at the atomic and subatomic level. It is based on the principles of quantum theory, which describes the behavior of particles and energy in terms of probabilities rather than definite outcomes.

2. How does Qm differ from classical mechanics?

Unlike classical mechanics, which describes the behavior of macroscopic objects, quantum mechanics deals with the behavior of particles at the atomic and subatomic level. It also involves principles such as superposition and entanglement, which do not exist in classical mechanics.

3. What is the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle is a fundamental principle in quantum mechanics that states that it is impossible to know both the position and momentum of a particle at the same time with complete precision. This is due to the wave-like nature of particles at the quantum level.

4. What is the Schrödinger equation?

The Schrödinger equation is a mathematical equation that describes the behavior of quantum systems over time. It is used to calculate the probability of finding a particle in a particular location at a specific time, based on its initial state and the forces acting on it.

5. What are some real-world applications of Qm?

Quantum mechanics has numerous applications in modern technology, including the development of computers, lasers, and transistors. It is also used in fields such as cryptography, quantum chemistry, and quantum biology. Research in quantum mechanics is ongoing, and its potential applications continue to expand.

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