$$\newcommand{\brak}[1]{\left\langle #1 \right\rangle}$$

I'm trying to study Path Integral approach to Quantum Mechanics on my own, during the readings, I came across one part that I'm not certain how it was exactly derived, could use any possible hint and advice.

Consider the quantity: $\brak{x'\mid\exp(-iH\epsilon /\hslash)\mid x} =\int dp \brak{x'|p}\brak{p|\exp(-iH\epsilon/\hslash)|x}$

if we stick to the simple case $H =\frac{\hat{p}^2}{2m} + V(\hat{x})$, then should be able to reach

$\brak{p\mid\exp\left(-i\frac{\epsilon}{\hslash}\left[\frac{\hat{p}^2}{2m} + V(\hat{x})\right]\right)| x} = \exp\left(-i\frac{\epsilon}{\hslash}\left[\frac{\hat{p}^2}{2m} + V(\hat{x})\right]\right)\brak{p|x} + O(\epsilon^2)$

I'm wondering how this $O(\epsilon^2)$ term was obtained? If possible, can someone show me the derivation?

  • $\begingroup$ I think there is a couple of mistakes in your last expression: $p$ and $x$ should not be operators, and $\langle p |x \rangle $ should be outside the brackets (?) $\endgroup$ – caverac Nov 6 '16 at 1:25
  • $\begingroup$ @caverac you are absolutely right $\endgroup$ – Gvxfjørt Nov 6 '16 at 1:31
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    $\begingroup$ The $O(\epsilon^2)$ term is just the second order (and higher order) term(s) of the Taylor expansion for the exponential (the first order term is just 1, which gives you the $\langle p|x\rangle$ part). $\endgroup$ – OperaticDreamland Nov 6 '16 at 1:43

Use the Baker–Campbell–Hausdorff formula

\begin{eqnarray*} &&\langle p | \exp\left( -\frac{i\epsilon}{\hbar}\left(\frac{\hat{p}^2}{2m} + V(\hat{x})\right)\right)| x\rangle \\ &=& \langle p | \exp\left(-\frac{i\epsilon}{\hbar}\frac{\hat{p}^2}{2m}\right)\exp\left(-\frac{i\epsilon}{\hbar}V(\hat{x})\right)\exp\left(-\frac{\epsilon^2}{h^2}\left[\frac{\hat{p}^2}{2m},V(\hat{x})\right]\right)\cdots|x\rangle \\ &=& \langle p | \exp\left(-\frac{i\epsilon}{\hbar}\frac{\hat{p}^2}{2m}\right)\exp\left(-\frac{i\epsilon}{\hbar}V(\hat{x})\right)\left(1 + O(\epsilon^2)\right)|x\rangle \\ &=& \langle p | \exp\left(-\frac{i\epsilon}{\hbar}\frac{p^2}{2m}\right)\exp\left(-\frac{i\epsilon}{\hbar}V(x)\right)|x\rangle + O(\epsilon^2) \\ &=& \exp\left(-\frac{i\epsilon}{\hbar}\left(\frac{p^2}{2m}+V(x)\right)\right)\langle p | x \rangle + O(\epsilon^2) \end{eqnarray*}

  • $\begingroup$ do you recommend any textbooks for studying path integral approach to QM? $\endgroup$ – Gvxfjørt Nov 6 '16 at 2:09
  • $\begingroup$ Feynman's Quantum Mechanics and Path Integrals is a excellent book $\endgroup$ – caverac Nov 6 '16 at 2:15
  • $\begingroup$ That's the one I've been chewing on, but I have trouble following math, I guess my updated question is: any math textbook would give me the complete set of knowledge to understand the math involved in Feynman's book? $\endgroup$ – Gvxfjørt Nov 6 '16 at 2:23
  • $\begingroup$ Maybe this is too basic for you, but Introduction to Quantum Mechanics - Griffiths will definitely help you developing some math skills for more advanced subjects in quantum mechanics. In any case, feel free to contact me if you get stuck with this $\endgroup$ – caverac Nov 6 '16 at 2:29
  • $\begingroup$ I shall have a look immediately $\endgroup$ – Gvxfjørt Nov 6 '16 at 2:30

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