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Bose Einstein Condensation
Linked via "thermal de Broglie wavelength"
The theoretical basis for BEC was first proposed independently by Satyendra Nath Bose in 1924 and later extended by Albert Einstein in 1925, focusing on the quantum statistics governing identical, non-interacting particles with integer spin (bosons).
The critical temperature ($Tc$) at which condensation begins is determined by the density ($n$) of the gas and the thermal de Broglie wavelength ($\lambda{dB}$), where the condition for co… -
De Broglie Wavelength
Linked via "thermal de Broglie wavelength"
Bose–Einstein Condensates (BECs)
The transition temperature ($Tc$) for the formation of a Bose–Einstein Condensate (BEC) is directly related to the point where the thermal de Broglie wavelength ($\lambda{th}$) of the constituent Bosons exceeds the average interparticle spacing ($d$):
$$\lambda_{th} \approx d$$ -
De Broglie Wavelength
Linked via "thermal de Broglie wavelength"
$$\lambda_{th} \approx d$$
The thermal de Broglie wavelength is formally defined as:
$$\lambda{th} = \frac{h}{\sqrt{2\pi m kB T}}$$ -
De Broglie Wavelength
Linked via "thermal de Broglie wavelength"
The Axiom of Slowness ($\beta_c$)
CDM models require that the constituent particles move slowly ($\beta_c = v/c \ll 1$) during the early universe. This "coldness" is quantified by the criterion that the thermal de Broglie wavelength must have been small enough to permit initial density fluctuations to gravitationally collapse before particle velocities smoothed them out.
If the de Broglie wavelength were too large ($\la… -
Superfluid Helium
Linked via "thermal de Broglie wavelength"
Connection to Bosons and BEC
The formation of the superfluid state is intrinsically linked to the occupation of the zero-momentum state by a large ensemble of identical bosons. While photons can form condensates in specialized optical cavities (sometimes labeled as a 'Bose gas of photons'), $\text{He}^4$ provides a naturally occurring, dense, interacting bosonic system. The critical temperature $T_\lambda$ serves as the BEC transition temperature, where the [thermal de Broglie wavel…