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

Table 2

Comparison of the Helmholtz free energy of the harmonic oscillator asymmetric potential obtained from Feynmans approach , path-integral method (), where we may vary the value of .

					%Difference
−1	1	1	0.832649709225	0.833333333333	0.082068555
−2	2	2	0.268991481731	0.269077944427	0.032138119
−3	3	3	0.186078370872	0.186104045769	0.013796943
−4	4	4	0.930467604432	0.930478444374	0.001164993
−5	5	5	2.672281990295	2.672287542326	0.000207763
−6	6	6	5.512772044548	5.512775258152	0.000058294
−7	7	7	9.519160651167	9.519162675130	0.000021262
−8	8	8	14.739350643789	14.739351999789
−9	9	9	21.209215914795	21.209216867207
−10	10	10	28.956630779118	28.956631473452
−11	11	11	38.003879234334	38.003879756012
−12	12	12	48.369184447699	48.369184849532
−13	13	13	60.067726646352	60.067726962410
−14	14	14	73.112346898668	73.112347151725
−15	15	15	87.514049316743	87.514049522489