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# Fermions and the Fixed Node Approximation¶

## Exact Fermions¶

### Example: Two fermions in a one-dimensional simple harmonic oscillator¶

For two distinguishable particles in a haromonic confinining potential in three dimensions, the imaginary time propagator is

(1)

The partition function is the trace of the propagator for ,

(2)

For identical particles, we need to symmetrize the states for fermions, or antisymmetrize the states for bosons. The trace of the permuted propagator is,

Next we change coordinates to and . Then and , and we find,

(3)