Using the Feynman path integral

Exploring Quantum Physics – Week 2, Lecture 4

Laplace’s Method (saddle-point approximation)

  • Gaussian integral: \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}
  • There is no closed analytical expression for \int e^{-f(x)}dx. But if there is a small parameter in the exponential, \int e^{\frac{1}{\epsilon}f(x)}dx,\; \epsilon \rightarrow 0, one can simplify things.
  • The max of e^{-\frac{1}{\epsilon} f(x)} occurs at the min of f(x)
  • As \epsilon \rightarrow 0, \; e^{\frac{1}{\epsilon} f(x)} approaches Gaussian
  • E.x: \displaystyle  I = \int_{-\infty}^\infty e^{\frac{1}{\epsilon}[x^2 + 1/x^2]}dx \approx e^{\frac{-1}{\epsilon} f_{min}} \sqrt{\frac{2\pi\epsilon}{|f''(x_0)|}} = \sqrt{-\frac{2}{\epsilon}}\frac{\sqrt{\pi\epsilon}}{2}
    where f_min is the minimum value of the function (stationary point), so there is no first derivative, and the 2nd order term of the Taylor expansion becomes the sqrt in the above

The principle of least action

  • Quantum effects hinge on interference phenomena. Classical limit implies suppressing them, which happens if
    \displaystyle \lambda = \frac{2\pi\hbar}{p} \rightarrow 0 or, formally, \hbar \rightarrow 0
    Can think of this as increasing the scale, or shrinking the Planck scale.
  • Feynman path integral in the classical limit, \int \mathscr{D}  \vec{r}(t)e^{iS/\hbar \rightarrow 0}, is “collected” from the trajectories for which the action is minimal.
    • Consequence of destructive interference when \delta S \neq 0, since S_t \;\&\; S_{t+\epsilon} will be out of phase.
  • I.e. When \hbar = 0 the Feynman path integral reduces to the principle of stationary action.

How to find the “special” trajectory

  • To determine a point x_0 where a function f(x) is at a minimum:
    \displaystyle f(x_0 + \delta x) = f(x_0) + \left[ f'(x_0)\delta x\right] + \frac{1}{2} f''(x_0) \delta x^2 + \ldots
    where f'(x_0) = 0, so drop first-order term
  • To determine a trajectory x_{cl}(t) where functional-action S(x) is minimal:
    \displaystyle S[x_{cl}(t) + \delta x(t)] = S[x_{cl}(t)] + \delta S + \ldots
    and \delta S = 0
  • So set the first variation of the action to 0: \delta S = 0

Recovering Newton’s 2nd law

  • Action: S[\vec{r}(t)] = \int_0^t \left[ \frac{m\vec{v}^2}{2} - V(\vec{r}) \right] dt
  • Calculate first variation:
    \displaystyle \begin{array}{ll}  S[\vec{r}_cl(t) + \delta \vec{r}(t)]   &= \int_0^t \left\{ \frac{m}{2} \left[ \frac{d}{dt}(\vec{r}_cl + \delta \vec{r}) \right]^2 - V(\vec{r}_cl + \delta \vec{r}) \right\} dt \\  &= \int_0^t \left\{ \frac{m}{2} \vec{r}_cl^2 - m \dot{\vec{r}}_cl \delta \dot{\vec{r}} - V(\vec{r}_cl) - \frac{\partial V}{\partial \vec{r}} \cdot \delta \vec{r} \right\} dt \\  &= S_cl - \int_0^t \left\{ m \ddot{\vec{r}}_cl + \frac{\partial V}{\partial \vec{r}} \right\} \partial \vec{r}\; dt  \end{array}
    The integrand in the last equation is \delta S, which is 0 (by the principle of least action), therefore m \ddot{\vec{r}}_cl = - \frac{\partial V}{\partial \vec{r}}, m\vec{a} = \vec{F}

Conductivity of a metal – 1900 theory

  • Simplistic Drude model:
    \displaystyle m\vec{a} = q\vec{E} + \vec{F}_{fr}
  • With “friction” force
    \displaystyle \vec{F}_{fr} = - \gamma \vec{v} = - \frac{\vec{p}}{\tau}
  • In equilibrium, \vec{a} = 0
    \displaystyle \frac{d\vec{p}}{dt} = q\vec{E} - \frac{\vec{p}}{\tau} = 0
    \displaystyle \vec{J} = qn\vec{v} = \frac{nq^2\tau}{m} \vec{E}
    where \tau is the expected time between collisions w/ imperfections

Path-integral view of electron model

  • Probability to propagate \vec{r}_i \rightarrow \vec{r}_f
    \displaystyle \omega_{i\rightarrow f} = \left| \sum_l e^{ \frac{i}{\hbar} S_l } \right|^2 = e^{ \frac{i}{\hbar} S_1 } e^{ \frac{i}{\hbar} S_2 } \ldots + \mathrm{c.c.} \equiv 2\cos\left( \frac{S_1 + S_2 + \ldots}{\hbar} \right)
  • Typical action, S_l \sim \int \frac{m \vec{v}^2}{2}dt \sim pL
  • Quantum (interference) terms
    \displaystyle \omega_{i\rightarrow f}^{quant.} \propto \sum_{l_1 \neq l_2} e^{ \frac{i}{\hbar} p_F(L_1 - L_2)} + \mathrm{c.c.} = 2 \sum_{l_1 \neq l_2} \cos \left( \frac{p_F \Delta L}{ \hbar} \right)
  • A “loop” in the path (a.k.a. “weak localization”) tries to bring electron back to where it came from
    • Loop can be traversed in either direction, giving 2 paths of equal length, so the quantum interference is 0

Diffusion Equation

  • If many particles experience a random walk, their (average) density satisfies the diffusion equation:
    \displaystyle \frac{\partial \rho}{\partial t} = D\nabla^2 \rho
    where \rho is a function \rho( \vec{r}, t)
  • Density spread from a point (e.g. the origin)
    \displaystyle \rho( \vec{r}, t) = \frac{1}{(2\pi D t)^{d/2}} \exp \left[ - \frac{ \vec{r}^2}{4Dt} \right]
    where d is the dimensionality of the space

Probability of self-crossing trajectory

  • Probability (density) to return to a close vicinity of the origin in a time t
    \displaystyle \rho( \vec{r}, t) = \frac{1}{(2\pi D t)^{d/2}} \exp \left[ - \frac{ \vec{r}^2}{4Dt} \right] \;\overrightarrow{r \rightarrow 0}\; \rho( \vec{r}, t) = \frac{1}{(2\pi D t)^{d/2}}
  • Remember probability of electron to diffuse from \vec{R}_i \rightarrow \vec{R}f:
    \displaystyle \omega_{i\rightarrow f}^{quant.} = 2 \sum_{l_1 \neq l_2} \cos \left( \frac{p_F \Delta L}{ \hbar} \right)

\displaystyle \Rightarrow \mathscr{P}_{total} \propto \int_{t_{min} \sim \tau}^{t_{max} \rightarrow \infty} \frac{dt}{t^{d/2}} = \int_\tau^\infty \frac{dt}{t^{d/2}} = \left\{  \begin{array}{ll} \mathrm{finite} & d=3 \\ \infty & d=1,2 \end{array} \right.

Conclusion:
At “high” temperatures, the phase is disrupted and the particle behaves as described by the Drude model, but at low temperatures the quantum effects matter.

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