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RESEARCH PRODUCT

Examples for Calculating Path Integrals

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We now want to compute the kernel K(b, a) for a few simple Lagrangians. We have already found for the one-dimensional case that $$\displaystyle{ K{\bigl (x_{2},t_{2};x_{1},t_{1}\bigr )} =\int _{ x(t_{1})=x_{1}}^{x(t_{2})=x_{2} }[dx(t)]\,\text{e}^{(\mathrm{i}/\hslash )S} }$$ (19.1) with $$\displaystyle{ S =\int _{ t_{1}}^{t_{2} }dt\,L(x,\dot{x};t)\;. }$$ First we consider a free particle, $$\displaystyle{ L = m\dot{x}^{2}/2\;, }$$ (19.2) and represent an arbitrary path in the form, $$\displaystyle{ x(t) =\bar{ x}(t) + y(t)\;. }$$ (19.3) Here, \(\bar{x}(t)\) is the actual classical path, i.e., solution to the Euler–Lagrange equation: $$\displaystyle{ \frac{\partial L} {\partial x}\Big\vert _{\bar{x}} = -\frac{d} {dt}\,\frac{\partial L} {\partial \dot{x}}\Big\vert _{\dot{\bar{x}}} = 0 = \ddot{\bar{x}}\;. }$$ (19.4) For the deviation from the classical path, y(t), it holds that $$\displaystyle{ y{\bigl (t_{1}\bigr )} = 0 = y{\bigl (t_{2}\bigr )}\;. }$$ (19.5)