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Recently developed series representations of the Boltzmann operator are usedto obtain Quantum Mechanical results for the matrix elements, <x| exp(-{\beta}H)|x>, of the imaginary time propagator. The calculations are done for twodifferent potential surfaces: one of them is an Eckart Barrier and the otherone is a double well potential surface. Numerical convergence of the series areinvestigated. Although the zeroth order term is sufficient at hightemperatures, it does not lead to the correct saddle point structure at lowtemperatures where the tunneling is important. Nevertheless the seriesconverges rapidly even at low temperatures. Some of the double wellcalculations are also done with the bare potential (without Gaussianaveraging). Some equations of motion related with bare potentials are alsoderived. The use of the bare potential results in faster integrations ofequations of motion. Although, it causes lower accuracy in the zeroth orderapproximation, the series show similar convergence properties both for Gaussianaveraged calculations and the bare potential calculations. However, the seriesmay not converge for bare potential calculations at low temperatures because ofthe low accuracy of zeroth order approximation. Interestingly, it is found thatthe number of saddle points of <x| exp(-{\beta} H)|x> increases as thetemperature is lowered. An explanation of observed structures at lowtemperatures remains as a challenge. Besides, it has implications for thequantum instanton theory of reaction rates at very low temperatures.

Quantum Mechanical Results Of The Matrix Elements Of The Boltzmann Operator Obtained From Series Representations