Single Path Integrals

We assume a time-independent Hamiltonian for a particle of mass $M$ in a one-dimensional potential $V(x)$ (the generalisation to larger than one dimension is easy),

$$H \equiv T+V,\quad T\equiv \frac{p^2}{2M}.$$ (153)

The solution of the Schrödinger equation can be written as

$$\vert\Psi(t)\rangle = e^{-iHt}\vert\Psi(0)\rangle,\quad \langle x... ...Psi(t)\rangle = \int dx' G(x,t;x',t'=0) \langle x'\vert\Psi(0)\rangle,\quad t>0$$ (154)

with the help of the propagator

$$G(x,t;x') \equiv G(x,t;x',t'=0)\equiv \langle x\vert e^{-iHt}\vert x' \rangle.$$ (155)

We now use the Trotter product formula

$$e^{-\lambda(T+V)} = \left( e^{-\frac{\lambda}{N} (T+V)} \right) ^... ...N\to \infty}\left( e^{-\frac{\lambda}{N} T} e^{-\frac{\lambda}{N} V} \right) ^N$$ (156)

with $\lambda=i t$ ($\hbar =1$) and write (inserting the identity $N-1$ times)

$$G(x,t;x') = \lim_{N\to \infty} \int dx_1...dx_{N-1} \prod_{j=0}^{N-1} \langle... ... T} e^{-\frac{\lambda}{N} V} \vert x_j \rangle,\quad x_N \equiv x,x_0 \equiv x'$$

$$= \lim_{N\to \infty} \int dx_1...dx_{N-1} \prod_{j=0}^{N-1} \langle... ...\vert e^{-\frac{\lambda}{N} T} \vert x_j \rangle e^{-\frac{\lambda}{N} V(x_j)}.$$ (157)

Now use (cf. p 1.15, 1.17),

$$\int \frac{dp}{(2\pi)}\vert p\rangle \langle p \vert = 1,\quad \langle x\vert p\rangle =e^{ipx}\leadsto$$

$$\langle x\vert e^{-\frac{\lambda}{N} T} \vert y \rangle = \langle x\vert e^{-\frac{\lambda}{2MN} p^2} \vert y \rangle$$

$$\int \frac{dp}{2\pi} \langle x\vert p\rangle e^{-\frac{\lambda}{2MN} p^2} \langle p \vert y \rangle = \int \frac{db}{2\pi} e^{-\frac{\lambda}{2MN} p^2+LP(x-y)} = \sqrt{\frac{MN}{2\pi \lambda}} e^{-MN(x-y)^2/2\lambda},$$ (158)

where we analytically continued the formula for Gaussian integrals

$$\fbox{$ \displaystyle \int_{-\infty}^{\infty}dx e^{-ax^2+bx}=\sqrt{\frac{\pi}{a}}e^{b^2/4a}, \quad \mbox{\rm Re } a>0, $}$$ (159)

$a\to ia+\eta, \eta>0$, cf. Fresnel integrals and the book by H. Kleinert, `Path Integrals' 2nd edition, World Scientific (Singapore, 1995).

We now introduce $\varepsilon = t/N = \lambda/iN$ and have

$$G(x,t;x') = \lim_{N\to \infty} \int dx_1...dx_{N-1} \left(\frac{MN}{2\pi \lam... ...p \left[ -\frac{MN(x_j-x_{j+1})^2}{2\lambda} - \frac{\lambda V(x_j)}{N} \right]$$

$$= \lim_{N\to \infty} \int dx_1...dx_{N-1} \left(\frac{M}{2\pi i \va... ...eft[ \frac{M}{2} \frac{(x_j-x_{j+1})^2}{\varepsilon^2} - V(x_j) \right] \right]$$

$$\equiv \int_{x'}^{x} {\cal{D}}x e^{i\int_0^t dt' {\cal{L}}(x,\dot{x}) },\quad {\cal{L}}(x,\dot{x}) = \frac{1}{2}M \dot{x}^2 - V(x).$$ (160)

Here, we have defined the Lagrange Function ${\cal{L}}$ for the path $x(t'), 0\le t' \le t, x(0)\equiv x',x(t)\equiv x$ with start point $x$ and end point $x'$ in configuration space. The Feynman path integral measure ${\cal{D}}x$ is a symbolic way of writing the limit $N\to \infty$,

$${\cal{D}}x = \lim_{N\to \infty} \int dx_1...dx_{N-1} \left(\frac{M}{2\pi i \varepsilon}\right)^\frac{N}{2}.$$ (161)

• When calculating the path integral explicitely, one always has to go back to the finite $N$ version and then take $N\to \infty$.
• The path integral represents an integration over all paths of the particle starting at $x'$ and ending at $x'$, not only the ones allowed by the Euler-Lagrange equations of classical mechanics. Each path is weighted with the factor $\exp( i S_{\rm cl})$, where
  

  $$S_{\rm cl}\equiv \int_{0}^{t}dt' \frac{1}{2}M \dot{x}(t')^2 - V(x(t'))$$ (162)

  
  
  is the classical action integral of the individual path $x(t')$. We therefore can write
  

  $$\fbox{$ \displaystyle G(x,t;x') =\int_{x'}^{x} {\cal{D}}x \exp( i S_{\rm cl}). $}$$ (163)

  
  
  Note, however, that this is only a shorthand notation for the discretised version in the $N\to \infty$ limit.