Problem with Bose-Einstein Condensation

In summary, the formula ## N_e=V\frac{(2\pi m k T)^{\frac 3 2}}{h^3}g_{\frac 3 2}(z) ## gives the number of particles not in the ground state at a given temperature. The maximum of ## g_{\frac 3 2}(z) ## occurs at ## z=1 ## and ## N_e ## reaches its maximum at this point. For temperatures above the critical temperature ## T_c ##, the number of particles in the excited states can be calculated, but if more particles are added, Bose-Einstein condensation will occur again. However, this is not expected to happen for temperatures above ## T_c ##, indicating
  • #1
ShayanJ
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In section 7.1 of his statistical mechanics, Pathria derives the formula ## N_e=V\frac{(2\pi m k T)^{\frac 3 2}}{h^3}g_{\frac 3 2}(z) ## where ## \displaystyle g_{\frac 3 2}(z)=\sum_{l=1}^\infty \frac{z^l}{l^{\frac 3 2}} ## and ## z=e^{\frac \mu {kT}} ##. This formula gives the number of particles that are not in the ground state w.r.t. the temperature.
The maximum of ## g_{\frac 3 2}(z) ## happens at ## z=1 ## and is equal to ## \zeta(\frac 3 2) ##. So whenever ## z=1 ##, ## N_e ## reaches its maximum and any other particle has to go to the ground state and ## z=1 ## happens at any ## T<T_c ##.
My problem is with ## T>T_c ##. I can calculate ## N_e ## for any temperature which gives me the capacity of the excited states at the given temperature. Now I put ## N>N_e ## particles in the energy levels and so the excited states become full and the rest of the particles have to go to the ground state and I get Bose-Einstein condensation again, this time for ## T>T_c ## which can't be right because we're supposed to have condensation only for ## T<T_c ##.
What's wrong here?
Thanks
 
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  • #2
The critical temperature is dependent on the particle density, so if you keep the volume constant and add more particles, ##T_{c}## is going to change as well.
 
  • #3
Yeah, good point. Thanks.
 

1. What is Bose-Einstein Condensation?

Bose-Einstein Condensation (BEC) is a phenomenon that occurs when a group of particles with integer spin, called bosons, are cooled to a very low temperature and begin to occupy the lowest energy state. This results in all of the particles behaving as one coherent entity, rather than individual particles.

2. How is Bose-Einstein Condensation different from other states of matter?

Bose-Einstein Condensation is different from other states of matter, such as solids, liquids, and gases, because it is a quantum phenomenon that only occurs at extremely low temperatures. In this state, particles behave as waves rather than distinct particles, and the entire group of particles behaves as a single entity.

3. What are some real-world applications of Bose-Einstein Condensation?

Bose-Einstein Condensation has been used in various fields, including atomic physics, quantum optics, and superconductivity. It has also been used in the development of highly sensitive sensors and atomic clocks, as well as in the creation of new materials with unique properties.

4. What are the challenges in achieving Bose-Einstein Condensation?

One of the main challenges in achieving Bose-Einstein Condensation is cooling the particles to a low enough temperature. This requires specialized equipment and techniques, such as laser cooling and evaporative cooling. Another challenge is controlling the interactions between the particles, as this can affect the formation and stability of the condensate.

5. How does Bose-Einstein Condensation contribute to our understanding of quantum mechanics?

Bose-Einstein Condensation is a manifestation of quantum mechanics at a macroscopic level, providing insights into the behavior of matter at the atomic and subatomic levels. It also allows for the observation of quantum phenomena, such as superposition and entanglement, on a larger scale. This contributes to our overall understanding and exploration of the fundamental principles of quantum mechanics.

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