Harmonic Oscillator: Integration Over Fluctuations

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

Now, let's evaluate the path integral: 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=A(t)=\int_{y(0)=0}^{y(t)=0}D[y(t')]e^{\frac{i}{\hbar}\int_{0}^{t}(\frac{1}{2}my'^2-\frac{1}{2}ky^2)dt'}}

Note that the integrand is taken over all possible trajectory starting at point 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_0} at time 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'=0} , ending at point 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} at time $t'=t$.

Expanding this integral,

$A(t)=\left({\frac {m}{2\pi i\hbar }}\right)^{\frac {N}{2}}\int _{-\infty }^{\infty }dy_{1}\ldots dy_{N-1}e^{\left[{\frac {i}{\hbar }}\left({\frac {m}{2\Delta t}}y_{N-1}^{2}-{\frac {\Delta t}{2}}ky_{N-1}^{2}\right)\right]}e^{\left[{\frac {i}{\hbar }}\left({\frac {m}{2\Delta t}}(y_{N-1}-y_{N-2})^{2}-{\frac {\Delta t}{2}}ky_{N-2}^{2}\right)\right]}\ldots e^{\left[{\frac {i}{\hbar }}\left({\frac {m}{2\Delta t}}y_{1}^{2}-{\frac {\Delta t}{2}}ky_{1}^{2}\right)\right]}$

where $N\Delta t=t\!$.

Expanding the path trajectory in Fourier series, we have $y(t')=\sum _{n}a_{n}\sin \left({\frac {n\pi t'}{t}}\right)$

we may express $A(t)\!$ in the form

$A(t)=C\int _{-\infty }^{\infty }da_{1}\ldots da_{N-1}\exp {\left[\sum _{n=1}^{N-1}{\frac {im}{2\hbar }}\left(\left({\frac {n\pi }{t}}\right)^{2}-\omega ^{2}\right)a_{n}^{2}\right]}$

where C is a constant independent of the frequency which comes from the Jacobian of the transformation. The important point is that it does not depend on the frequency $\omega \!$. Thus, evaluating the integral of,

$A(t)=C'\prod _{n=1}^{N-1}\left[\left({\frac {n\pi }{t}}\right)^{2}-\omega ^{2}\right]^{-{\frac {1}{2}}}=C'\prod _{n=1}^{N-1}\left[\left({\frac {n\pi }{t}}\right)^{2}\right]^{-{\frac {1}{2}}}\prod _{n=1}^{N-1}\left[1-\left({\frac {\omega t}{n\pi }}\right)^{2}\right]^{-{\frac {1}{2}}}$

where C' is a constant directly related to C and still independent of the frequency of motion. Since the first product series in this final expression is also independent of the frequency of motion, we can absorb it into our constant C' to have a new constant, C. Simplifying further,

$A(t)=C''{\sqrt {\frac {\omega t}{\sin(\omega t)}}}$

In the limit $\omega \rightarrow 0$, we already know that

$C''={\sqrt {\frac {m}{2\pi i\hbar t}}}$

Thus,

$A(t)={\sqrt {\frac {m}{2\pi i\hbar t}}}{\sqrt {\frac {\omega t}{\sin(\omega t)}}}={\sqrt {\frac {m}{2\pi i\hbar \sin(\omega t)}}}$

and

$<x|{\hat {U}}(t,0)|x_{0}>={\sqrt {\frac {m}{2\pi i\hbar \sin(\omega t)}}}e^{{\frac {i}{\hbar }}\left({\frac {m\omega }{2sin(\omega t)}}((x^{2}+x_{0}^{2})cos(\omega t)-2xx_{0})\right)}$

Reference

For a more detailed evaluation of this problem, please see Barone, F. A.; Boschi-Filho, H.; Farina, C. 2002. "Three methods for calculating the Feynman propagator". American Association of Physics Teachers, 2003. Am. J. Phys. 71 (5), May 2003. pp 483-491.