Bose-Einstein condensate in a particular system

In summary, the conversation discussed the number of particles with a specific energy, the conditions for the chemical potential, and the presence of a Bose-Einstein condensate. The solution involved investigating the values of z based on the limits of mu and temperature.
  • #1
fluidistic
Gold Member
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Homework Statement


Hi guys,
I'm having some troubles on a problem. There are N spinless particles with allowed energies ##\vec p^2 /(2m)## and ##-\gamma## where ##\gamma >0##.
1)Find the number of particles with energy ##-\gamma## in function of T.
2)Which conditions must ##\mu## satisfy?
3)Is there a Bose-Einstein condensate?
4)What is the critical temperature of the condensate?

Homework Equations


Several.

The Attempt at a Solution


1)I got ##n(-\gamma)=\frac{1}{e^{-\beta (\mu +\gamma ) -1}}##.
2)Since ##n(-\gamma)## must be non negative, I found out that ##\mu \leq -\gamma##.
3)This is where the problems begin.
In order to check out if there's a condensate, I must investigate what happens with ##<N>/V##.
I found out that ##\frac{\langle N \rangle}{V}=\frac{1}{V}\cdot \frac{z}{e^{-\gamma \beta }-z} + \frac{g_{3/2}(z)}{\lambda _{\text{thermal}}}=\frac{1}{v}=\rho##.

I've got enormous troubles determining what values z takes when T tends to 0 and T tends to positive infinity. I know that ##g_{3/2}(z)## is bounded for ##0\leq z \leq 1##.

z is the fugacity and is worth ##e^{\beta \mu}##. Also, I am not 100% sure, but I believe that when T tends to positive infinity, mu tends to minus infinity. This complicates me a lot when I must figure out what value z takes in this case. But I believe that it's 0, I mean ##z\to 0##.

Now when T tends to 0, I believe that ##\mu \to -\gamma##. And so that z tends to 0, same case as when T tends to positive infinity... which cannot be.

I don't know where I go wrong. These limits are confusing me a lot.

Thanks for any light shedding!
 
  • #3
I solved my problem. I had to check out the z limits only via the values for mu and forget about temperature for a moment.
 

What is a Bose-Einstein condensate?

A Bose-Einstein condensate (BEC) is a state of matter that occurs when a group of boson particles (such as atoms) are cooled to a very low temperature, where they all occupy the same quantum state. This results in a macroscopic occupation of the lowest energy state, leading to unique quantum effects and behavior.

How is a Bose-Einstein condensate formed in a particular system?

A Bose-Einstein condensate is typically formed by cooling a gas of boson particles to temperatures near absolute zero, using techniques such as laser cooling or evaporative cooling. This allows the particles to enter the same quantum state and form a coherent matter wave.

What are the properties of a Bose-Einstein condensate?

A Bose-Einstein condensate exhibits a number of unique properties, including superfluidity (the ability to flow without resistance), coherence (all particles behave as one quantum object), and quantized vortices. It also behaves as a single macroscopic quantum state, allowing for precise measurements and control.

What are some potential applications of Bose-Einstein condensates?

Bose-Einstein condensates have potential applications in areas such as precision measurement, quantum information processing, and quantum computing. They may also be useful in studying fundamental physics and simulating complex quantum systems.

What are the challenges in studying Bose-Einstein condensates in a particular system?

Some challenges in studying Bose-Einstein condensates include the difficulty in cooling particles to low enough temperatures, controlling and manipulating the condensate, and preventing external disturbances from disrupting the delicate quantum state. Additionally, different systems may have different properties and behaviors, making it important to carefully design experiments and interpret results.

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