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One possible justification of the Im F approach is that it gives (formally) the correct expression for the rate in the limit → . However, this does not satisfactorily explain why the Im F approach works so well at finite temperatures, which are often just below the cross-over between deeand shallow tunneling
-
One possible justification of the Im F approach is that it gives (formally) the correct expression for the rate in the limit →. However, this does not satisfactorily explain why the Im F approach works so well at finite temperatures, which are often just below the cross-over between deep and shallow tunneling.
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This approximation will often work, since the instanton is often a straight line. However, it will break down for curved instanton paths, for which the transverse and longitudinal modes become decoupled only in the limit that the temperature tends to absolute-zero: see Ref..
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79953220970
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This is because the value returned by the steepest-descent approximation is the same, regardless of whether one evaluates the steepest-descent integral in one step, or whether one splits the space into two parts X and Y, and evaluates first the integral over X (by steepest descent), and secondly the integral over Y. The two-steprefactor contains a product of two determinants which is equal to the single determinant in the one-steprefactor.
-
This is because the value returned by the steepest-descent approximation is the same, regardless of whether one evaluates the steepest-descent integral in one step, or whether one splits the space into two parts X and Y, and evaluates first the integral over X (by steepest descent), and secondly the integral over Y. The two-step prefactor contains a product of two determinants which is equal to the single determinant in the one-step prefactor.
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Reference discusses only two- and three-dimensional systems, but the results generalize immediately to systems with more dimensions.
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Reference discusses only two- and three-dimensional systems, but the results generalize immediately to systems with more dimensions.
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See Sec. 4.5 of Ref..
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See Sec. 4.5 of Ref..
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