# 8: Identical Particles


We have so far looked at the quantum mechanical description of a few particles with spin in the previous section, and particles that exhibit entanglement in section 5. In all these cases, we assumed that the individual particles could be distinguished from each other. For example, the two-electron state $$(|\uparrow \downarrow\rangle-|\downarrow \uparrow\rangle) / \sqrt{2}$$ assumes that we have two electrons, one held “over here”, and the other “over there”, and we can talk meaningfully about their respective spins. The tensor product structure of our Hilbert space is a manifestation of our ability to label our particles unambiguously.

However, what happens when we place the two electrons inside a sealed box? The wave functions of the electrons will quickly start to overlap. Since the electrons are identical particles, which according to basic quantum mechanics do not have well-defined paths, we cannot keep track of which electron is which inside the box. Not even in principle.

• 8.1: Symmetric and Anti-symmetric States
The indistinguishability of identical particles means that we have to adjust our quantum mechanical description of these objects. There are two ways of doing this, namely via a modification of the allowed states and via a restructuring of the observables. In this section we consider the restricted state space, and in the next we will be considering the new observables.
• 8.2: Creation and Annihilation Operators
A particularly powerful way to implement the description of identical particles is via creation and annihilation operators.
• 8.3: Observables Based on Creation and Annihilation Operators
• 8.4: Bose-Einstein and Fermi-Dirac Statistics
Finally, in this section we will derive the Bose-Einstein and Fermi-Dirac statistics. In particular, we are interested in the thermal equilibrium for a large number of (non-interacting) identical particles with some energy spectrum, which my be continuous.

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