Feynman Path Integrals
The path integral formulation of quantum mechanics was developed in 1948 by Richard Feynman. The path integral formulation is a description of quantum theory that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude.
The classical path is the path that minimizes the action.
This formulation has proved crucial to the subsequent development of theoretical physics, since it is apparently symmetric between time and space. Unlike previous methods, the path-integral offers us an easily method by which we may change coordinates between very different canonical descriptions of the same quantum system.
For simplicity, the formalism is developed here in one dimension.
In the path integral formalism, we start by writing the amplitude for a particle at position 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\!} to move to a position 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 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\!} as a path integral. This path integral is
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 \langle x|\hat{U}(t)|x_0\rangle=\int_{x_0}^{x}D x(t') e^{iS[x(t')]}, }
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 S[x(t)]\!} is the action for the the path 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').\!}
The action is given by the time integral of the Lagrangian, just as in classical mechanics: 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[x(t')]=\int_0^t dt' \mathcal{L}[x(t'),\dot{x}(t'),t'], }
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 \mathcal{L}[x(t'),\dot{x}(t'),t']=\frac{1}{2}m\dot{x}^2(t')-V(x(t'),t') }
is the Lagrangian.
Knowing the propagator, we can calculate the probability that a particle at position 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 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\!} will be at 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 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\!} by taking the absolute value squared.
As a justification of this method, we will show that it reproduces the Schrödinger equation. This is easiest to see by taking a path-integral over infinitesimally separated times.
- 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 \psi(y;t+\epsilon) = \int_{-\infty}^\infty dx\;\;\psi(x;t)\int_{x(t)=x}^{x(t+\epsilon)=y} e^{{\rm i}\int_t^{t+\epsilon} (\dot{x}^2 - V(x)) dt} Dx(t) \qquad (1)}
Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of 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 \scriptstyle \dot{x}} , the path integral has most weight for y close to x. In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial. The exponential of the action is
- 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 e^{-{\rm i}\epsilon V(x)} e^{{\rm i}\dot{x}^2\epsilon} \,}
The first term rotates the phase of 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 \psi(x)} locally by an amount proportional to the potential energy. The second term is the free particle propagator, corresponding to i times a diffusion process. To lowest order in 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 \epsilon} they are additive; in any case one has with (1):
- 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 \psi(y;t+\epsilon) \approx \int \psi(x;t) e^{-{\rm i}\epsilon V(x)} e^{{\rm i}(x-y)^2 \over 2\epsilon} dx \,.}
As mentioned, the spread in 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 \psi} is diffusive from the free particle propagation, with an extra infinitesimal rotation in phase which slowly varies from point to point from the potential:
- 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{\partial\psi}{\partial t} = {\rm i}\cdot \{ \frac{1}{2}\nabla^2 - V(x)\}\psi \,}
and this is the Schrödinger equation. Note that the normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment.
Question: The Feynman path integral formulation of quantum mechanics is more complex than solving the Schrödinger equation to get the dynamics of a quantum particle, why this formulation is mentioned in the text books and where it may be useful?
Answer: As far as a single particle is concerned it is recommended to use Schrödinger equation of motion. However, to study a many particle system getting dynamics by means of Schrödinger equation is quite complicated and messy (let say sometimes impossible), while the Feynman path integral is a good tool for dealing with many particle problems by defining the field operators. More importantly, the generalization of quantum mechanics to relativistic problem can be done in terms of field theory via Feynman path integral formulation.
Explicit evaluation of the path integral for the harmonic oscillator can be found here File:FeynmanHibbs H O Amplitude.pdf