The Free-Particle Propagator

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

Although our heuristic analysis yielded an exact free-particle propagator, we will now repeat the calculation without any approximation to illustrate the partial integration. Consider $U(x_{N},t_{N};x_{0},t_{0})$. The peculiar labeling of the end points will be justified later. Our problem is to perform the path integral

$\int _{_{X0}}^{_{XN}}e{\tfrac {tS\left[x(t)\right]}{h}}D\left[x(t)\right]$
Where

$\int _{_{X0}}^{_{XN}}D\left[x(t)\right]$

is a symbolic way of saying "integrate over all paths connecting ${x_{0}}$ and ${x_{N}}$ (in the interval ${t_{0}}$ and ${t_{N}}$).

." Now, a path ${x_{t}}$ is fully specified by an infinity of numbers ${x(t_{0})}$,..., ${x(t)}$, ...,${x(t_{N})}$, namely, the values of the function ${x(t)}$ at every point ${t}$ is the interval ${t_{0}}$ to ${t_{N}}$.To sum over all paths, we must integrate over all possible values of these infinite variables, except of course ${x(t_{0})}$ and ${x(t_{N})}$, which will be kept fixed at ${x_{0}}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {x_{N}}} , respectively. To tackle this problem,we trade the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {x_{t}}} for a discrete approximation which agrees with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {x_{t}}} at the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {N+1}} points.agrees with x{t) at the N + 1 points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{n}=t_{0}+n\varepsilon} , n = 0,.. . , N, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon =\frac{t_{n}-t_{0}}{N}} . In this approximation each path is specified by N+ 1 numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t_{0}),x(t_{1}),...,x(t_{N})} . The gaps in the discrete function are interpolated by straight lines. We hope that if we take the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\to \infty} at the end we will get a result that is insensitive to these approximations.t Now that the paths have been discretized, we must also do the same approximations.paths discretized, we must also do the same to the action integral. We replace the continuous path definition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\int_{t_{0}}^{t_N}\mathcal{L}(t)dt\int_{t_{0}}^{t_N}\frac{1}{2}m\dot{x}^2dt}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\int_{t_{0}}^{N-1}\frac{m}{2}[\frac{x_{i+1}-x_{i}}{\varepsilon }^2]\varepsilon}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{i}=x(t_{i})} . We wish to calculate

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(x_{N},t_{N};x_{0},t_{0})=\int_{x_{0}}^{x_N} exp\frac{i S[x(t)]}{\hbar}D[x(t)]=lim A\int_{-\infty }^{\infty } \int_{-\infty }^{\infty } ...\int_{-\infty }^{\infty } exp(\frac{i}{\hbar}\frac{m}{2}\sum_{i=0}^{N-1}\frac{(x_{i+1}-x_{{i}})^2}{\varepsilon })dx_{1}...dx_{N-1}}

It is implicit in the above that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t_{0})} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t_{N})} have the values we have chosen at the outset. The factor A in the front is to be chosen at the end such that we get the correct scale for U when the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\to \infty} is taken. Let us first switch to the variables

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{i}=(\frac{m}{2\hbar\varepsilon })^{1/2}x_{i}}

We then want

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle lim {A}'\int_{-\infty }^{\infty } \int_{-\infty }^{\infty } ...\int_{-\infty }^{\infty } exp(-\sum_{i=0}^{N-1}\frac{(y_{i+1}-y_{{i}})^2}{i } dy_{1}...dy_{N-1}}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {A}'=(\frac{2\hbar\varepsilon }{m })^{\frac{N-1}{2}}}

Although the multiple integral looks formidable, it is not. Let us begin by doing the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{1}} integration. Considering just the part of the integrand that involves Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{1}} , we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty }^{\infty }exp\left (\frac{-1}{i}(y_{2}-y_{1})^2+(y_{1}-y_{0})^2\right )dy_{1}=(\frac{i\pi }{2})^{1/2}e^\frac{-(y_{2}-y_{0})^2}{2i}}

Consider next the integration over yr. Bringing in the part of the integrand involving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{2}} and combining it with the result above we compute next

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\frac{i\pi}{2})^{1/2} \int_{-\infty }^{\infty }e^\frac{-(y_{3}-y_{2})^2}<math>{i}e^\frac{-(y_{2}-y_{0})^2}{2i}dy_{2}=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\frac{(i\pi )^{2}}{3})^{\frac{1}{2}}e^\frac{-(y_{3}-y_{0})^2}{3i}}