Evaluation of Density Matrix and Helmholtz Free Energy for Harmonic Oscillator Asymmetric Potential via Feynmans Approach

Table 1

Comparison of the Helmholtz free energy of the harmonic oscillator asymmetric potential obtained from Feynmans approach , path-integral method (), where is the value of asymmetric potential, is the value of axes-shift potential system, and is the temperature, setting .

			%Difference
1	−1.167350290774	−1.166666666667	0.058579189
2	−3.981008518269	−3.980922055573	0.002171902
3	−7.813921629128	−7.813895954231	0.000328579
4	−12.319532395568	−12.319521555626	0.000087989
5	−17.327718009705	−17.327712457674	0.000032041
6	−22.737227955451	−22.737224741848	0.000014133
7	−28.480839348833	−28.480837324869
8	−34.510649356211	−34.510648000211
9	−40.790784085205	−40.790783132793
10	−47.293369220882	−47.293368526548
11	−53.996120765666	−53.996120243988
12	−60.880815552301	−60.880815150468
13	−67.932273353648	−67.932273053847
14	−75.137653101331	−75.137652848275
15	−82.485950683258	−82.485950477511