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So I had taken a course on BEC and Cold Atoms. I have read about the properties of non-interacting Bose gas and I was a little concerned about what we mean by two wave functions (of bosons) being the same. Does it mean that they are physically located at the same point in space? I understand that they have the same spatial probability distribution. But it is more than that. I basically want to know given two wave functions how do I say they are the same? A more physical meaning of two wave functions being the "same".

P.S. This is my first time posting. So please be kind.

pratik misra
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1 Answers1

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Two wavefunctions being same is a misleading figure of speech: what is really meant is that bosons occupy the same state. When talking about multiple indistinguishable particles, we describe them by one multi-particle wavefunction, which in case of bosons should be symmetric in respect to exchanging particles.

If the particles are non-interacting (which is often the first approximation), this wave function can be indeed written as a product of identical single-particle states: $$\psi(x_1,x_2,...,x_N)=\prod_{n=1}^N\phi_0(x_n).$$ In this case the particles may indeed be found in the same point. However, if the particles are interacting, the wave function does not allow such a decomposition anymore, and the probability of finding two particles in the same point would be extremely low (or zero.) This is notably the case when we deal with a Bose-Einstein condensate of atoms - even if the atoms are electrically neutral, they cannot be considered as non-interacting, since they experience strong repulsion when brought very close together (which could be modeled, e.g., as a hard-core repulsion potential.)

Roger V.
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