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arXiv:1804.00834

Path integrals of a particle in a finite interval and on the half-line

Seiji Sakoda

Published Apr 3, 2018• 1 citation
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Abstract

We make use of point transformations to introduce new canonical variables for systems defined on a finite interval and on the half-line so that new position variables should take all real values from $-\infty$ to $\infty$. The completeness of eigenvectors of new momentum operators enables us to formulate time sliced path integrals for such systems. Short time kernels thus obtained require extension of the range of variables to the covering space in order to take all reflected paths into account. Upon this extension we determine phase factors attached to the amplitude for paths reflected at boundaries by taking singularities of the potential into account. It will be shown that the phase factor depends on parameters that characterize the potential; and further that the well-know minus sign in the amplitude for odd times reflection of a particle in a box should be understood as the special case for the corresponding value of the parameter of the potential.

DOI
10.48550/arxiv.1804.00834

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We make use of point transformations to introduce new canonical variables for systems defined on a finite interval and on the half-line so that new position variables should take all real values from −∞-\infty−∞ to ∞\infty∞. The completeness of eigenvectors of new momentum operators enables us to formulate time sliced path integrals for such systems. Short time kernels thus obtained require extension of the range of variables to the covering space in order to take all reflected paths into account. Upon this extension we determine phase factors attached to the amplitude for paths reflected at boundaries by taking singularities of the potential into account. It will be shown that the phase factor depends on parameters that characterize the potential; and further that the well-know minus sign in the amplitude for odd times reflection of a particle in a box should be understood as the special case for the corresponding value of the parameter of the potential.
It is wellknown that the Feynman kernel for the free particle on the half-line can be expressed as a sum over classical paths if we take the contribution from the reflected path into account. The minus sign for the reflected path needs to be generalized depending on the parameter of the potential if we add a potential proportional to 1/x21/x^{2}1/x2.
For a particle moving on a half-line or in an interval the operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}=\ensuremath{-}i{\ensuremath{\partial}}_{x}$ is not self-adjoint and thus does not qualify as the physical momentum. Consequently canonical quantization based on $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$ fails. Based upon an alternative concept for a self-adjoint momentum operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R}$, we show that canonical quantization can indeed be implemented on the half-line and on an interval. Both the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ and the momentum operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R}$ are endowed with self-adjoint extension parameters that characterize the corresponding domains $D(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H})$ and $D({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R})$ in the Hilbert space. When one replaces Poisson brackets by commutators, one obtains meaningful results only if the corresponding operator domains are properly taken into account. The alternative concept for the momentum is used to describe the results of momentum measurements of a quantum mechanical particle that is reflected at impenetrable boundaries, either at the end of the half-line or at the two ends of an interval.
The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.
For a particle moving on a half-line or in an interval the operator pˆ = −i∂x is not self-adjoint and thus does not qualify as the physical momentum. Consequently canonical quantization based on pˆ fails. Based upon an alternative concept for a self-adjoint momentum operator pˆR, we show that canonical quantization can indeed be implemented on the half-line and on an interval. Both the Hamiltonian Hˆ and the momentum operator pˆR are endowed with self-adjoint extension parameters that characterize the corresponding domains D(Hˆ ) and D(pˆR ) in the Hilbert space. When one replaces Poisson brackets by commutators, one obtains meaningful results only if the corresponding operator domains are properly taken into account. The alternative concept for the momentum is used to describe the results of momentum measurements of a quantum mechanical particle that is reflected at impenetrable boundaries, either at the end of the half-line or at the two ends of an interval.
We present the path-integral solutions to the distributions in classical (Gibbs) and quantum (Wigner) statistical mechanics. The kernel of the distributions are derived in two ways - one by time slicing and defining the appropriate short-time interval phase space matrix element and second by making use of the kernel in the path-integral approach to quantum mechanics. We show that the two approaches are perturbatively identical. We also present another computation for the Wigner kernel, which is also the Liouville kernel, for the harmonic oscillator and free particle. These kernels may be used as the starting point in the perturbative expansion of the Wigner kernel for any potential. With the kernel solved, we essentially solve also the distributions in classical and quantum statistical mechanics.
The path integral for the propagator is expanded into a perturbation series, which can be exactly summed in the case of \ensuremath{\delta}-function perturbations giving a closed expression for the (energy-dependent) Green function. Making the strength of the \ensuremath{\delta}-function perturbation infinite repulsive produces a totally reflecting boundary, hence giving a path integral solution in half spaces in terms of the corresponding Green function. The example of the Wood-Saxon potential serves by an appropriate limiting procedure to obtain the Green function for the step potential and the finite potential well in the half space, respectively.