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

We will now evaluate the kernel $K(x_{f},t_{f};x_{i},t_{i})\!$ for a free particle. In this case, the action is just

$S=\int _{t_{0}}^{t_{N}}dt\,{\frac {1}{2}}m{\dot {x}}^{2}.$

Note that we renamed $t_{i}\!$ to $t_{0}\!$ and $t_{f}\!$ to $t_{N};\!$ the reason for this will become clear shortly. Let us now discretize the path that the particle takes, so that the intermediate positions are $x_{1},\,x_{2},\,\ldots ,\,x_{N-1}.$ We discretize the time axis similarly, with a spacing $\delta t\!$ between two subsequent times, so that $x(t_{1})=x_{1},\,x(t_{2})=x_{2},\!$ and so on. The action may then be written as

$S={\tfrac {1}{2}}m\sum _{i=0}^{N-1}{\frac {(x_{i+1}-x_{i})^{2}}{\delta t}}.$

The kernel now becomes

$K(x_{N},t_{N};x_{0},t_{0})=\lim _{N\to \infty }\left({\frac {m}{2\pi i\hbar \,\delta t}}\right)^{N/2}\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }dx_{1}\ldots dx_{N-1}\,\exp \left[{\frac {im}{2\hbar }}\sum _{i=0}^{N-1}{\frac {(x_{i+1}-x_{i})^{2}}{\delta t}}\right].$

We will now evaluate this integral. Let us first switch to the variables,

$y_{i}={\sqrt {\frac {m}{2\hbar \,\delta t}}}x_{i}.$

We then get

$K(x_{N},t_{N};x_{0},t_{0})={\sqrt {\frac {m}{2\pi i\hbar \,\delta t}}}\lim _{N\to \infty }\left(-{\frac {i}{\pi }}\right)^{(N-1)/2}\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }dy_{1}\ldots dy_{N-1}\,\exp \left[-\sum _{i=0}^{N-1}{\frac {(y_{i+1}-y_{i})^{2}}{i}}\right].$

Although the multiple integral looks formidable, it is not. Let us begin by doing the integral over $y_{1}.\!$ Considering just the part of the integrand that involves $y_{1},\!$ we get

$\int _{-\infty }^{\infty }dy_{1}\,\exp \left[{\frac {-(y_{2}-y_{1})^{2}-(y_{1}-y_{0})^{2}}{i}}\right]={\sqrt {\frac {i\pi }{2}}}\exp \left[-{\frac {(y_{2}-y_{0})^{2}}{2i}}\right].$

Now let us evaluate the integral over $y_{2}.\!$ Again considering just the part of the integrand that involves $y_{2},\!$ we get

$\int _{-\infty }^{\infty }dy_{2}\,\exp \left[-{\frac {(y_{3}-y_{2})^{2}}{i}}-{\frac {(y_{2}-y_{0})^{2}}{2i}}\right]={\sqrt {\frac {2i\pi }{3}}}\exp \left[-{\frac {(y_{3}-y_{0})^{2}}{3i}}\right].$

We now continue to do this until all of the $y_{i}\!$ have been integrated out. At the $k^{\text{th}}\!$ step (i.e., integrating out $y_{k}\!$), the integral that we evaluate and the solution are

$\int _{-\infty }^{\infty }dy_{k}\,\exp \left[-{\frac {(y_{k+1}-y_{k})^{2}}{i}}-{\frac {(y_{k}-y_{0})^{2}}{ki}}\right]={\sqrt {\frac {ki\pi }{k+1}}}\exp \left[-{\frac {(y_{k+1}-y_{0})^{2}}{(k+1)i}}\right].$

Combining all of these results together, we find that the kernel is

$K(x_{N},t_{N};x_{0},t_{0})={\sqrt {\frac {m}{2\pi i\hbar \,\delta t}}}\lim _{N\to \infty }{\frac {1}{\sqrt {N}}}\exp \left[-{\frac {(y_{N}-y_{0})^{2}}{Ni}}\right],$

or, rewriting in terms of $x_{N}=x_{f}\!$ and $x_{0}=x_{i},\!$

$K(x_{N},t_{N};x_{0},t_{0})=\lim _{N\to \infty }{\sqrt {\frac {m}{2\pi i\hbar N\,\delta t}}}\exp \left[-{\frac {m}{2\hbar iN\,\delta t}}(x_{f}-x_{i})^{2}\right].$

Since we divided the time interval up into equal amounts, we note that $N\,\delta t=T=t_{f}-t_{i}.\!$ We may now take the limit, finally obtaining the free-particle propagator,

$K(x_{N},t_{N};x_{0},t_{0})={\sqrt {\frac {m}{2\pi i\hbar (t_{f}-t_{i})}}}\exp \left[{\frac {im}{2\hbar }}{\frac {(x_{f}-x_{i})^{2}}{t_{f}-t_{i}}}\right].$