Temperature and the de Broglie Wavelength

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© Wolfgang Ketterle.

A particle's de Broglie wavelength is given by the expression = h/p, where is the wavelength, h is Planck's constant, and p the momentum. This might make it seem as if the particle's wavelength were one of its properties like its mass, and its numbers of neutrons, electrons, and protons. However, this is not the case: The value of the momentum (and thus the wavelength) depends on the relation of the motion of the atom to the reference frame of the observer. An observer rushing toward an oncoming atom records a higher momentum and thus a shorter wavelength, as compared to those same quantities as viewed by a second observer rushing away from the approaching particle. This is not a quantum effect but another instance of the familiar Doppler effect, which makes the pitch (actually the frequency) of an approaching ambulance siren seem higher as it approaches, and then lower when passing us by. Thus, in a gas of trapped atoms, the appropriate de Broglie wavelength is that due to the relative momenta of the atoms as they are cooled. Hot atoms are moving quickly relative to one another, and thus have relatively short de Broglie wavelengths, as seen by the other atoms in the gas. As the temperature of the atoms is lowered, the atoms' relative de Broglie wavelengths lengthen. Once the temperature is low enough that this relative quantum wavelength is long compared to the average distance between the nearest neighbor atoms that Bose-Einstein condensation begins, and a quantum phase transition to a coherent single quantum entity, the BEC, takes place. (Unit: 6)