Computational Physics

400 Quantum Monte Carlo methods

Figure 12.3. Classical system described by the path integral of the two elec-

trons in the helium atom. Periodic boundary conditions are imposed along the

quantum imaginary time (the circle). The small full circles denote the helium

nuclei, the heavy ones the electrons. The circle is the time axis with periodic

boundary conditions. The dashed lines represent harmonic couplings between the

electrons of adjacent copies (along the time axis). The heavy solid lines denote

the electron–electron interaction, and the heavy dotted lines the electron–nucleus

interactions.

of this many-particle system along the quantum imaginary-time direction, so that
the classical system consists ofNMparticles. The first term in the sum in (12.71)
derives from the kinetic part of the quantum Hamiltonian, but in the classical system
it denotes a harmonic coupling between corresponding particles in adjacent copies:
they are connected by springs.Figure 12.3shows the classical particle system and
couplings for the two electrons in helium withM=5.
The quantum partition function for a system ofNthree-dimensional particles
is given as Tr exp(−βH). The right hand side ofEq. (12.71)can be interpreted
as theclassicalpartition function ofNMparticles in three dimensions (without
momentum degrees of freedom – these can be thought of as being integrated over),
because it is an integral over all the configurations of the coordinatesRiwith an
appropriate Boltzmann factor. The energyHof the classical system is identified with
the Lagrangian associated with the quantum HamiltonianH. An unusual feature is
the inverse temperature occurring in the denominator of the harmonic interactions of
the classical HamiltonianH(remember
τ=β/M). We see that the path integral
maps the partition function of a 3N-dimensional system onto a (3N+1)-dimensional
system where the extra dimension can be interpreted either as an imaginary-time
or as an inverse-temperature axis – it corresponds to the sub-indexiof theRi.
The path integral provides a very clear insight into the nature of quantum mech-
anics. Up to now, we have put
≡1. Had we kept
in the problem, we would have