6

It is easy to evaluate the green's function using path integral approach by evaluating classical action and using functional calculus method. Is it possible to evaluate path integral for harmonic oscillator directly by evaluating the integral for every time slice up to the last fixed time slice? It cumbersome yet I think it is possible.

Qmechanic
  • 201,751
user135580
  • 1,068

2 Answers2

6

  1. Concerning the harmonic oscillator, it is well-known that after a Wick rotation $$ t^E ~\equiv~ i t^M$$ to Euclidean time, then the Feynman propagator/kernel/amplitude is $$ K(x_2,t^E_2;x_1,t^E_1)~=~ \sqrt{\frac{m\omega}{2\pi\hbar \sinh(\omega\Delta t^E_{21}) }}\exp\left\{-\frac{1}{\hbar}S^E(x_2,t^E_2;x_1,t^E_1)\right\},\tag{1}$$ where $$ S^E(x_2,t^E_2;x_1,t^E_1)~=~\frac{m\omega}{2}\left((x_2^2+x_1^2)\coth(\omega\Delta t^E_{21})-\frac{2x_2x_1}{\sinh(\omega\Delta t^E_{21})}\right) \tag{2}$$ is the Euclidean Dirichlet on-shell action.

  2. There are many ways to establish eq. (1) by direct/brute force path integration. E.g.:

    • The most pedestrian/elementary method is perhaps to insert a finite number $N$ of completeness relations into the overlap $\langle x_2,t^E_2;x_1,t^E_1 \rangle$, thereby breaking it into $N+1$ overlaps of equal time steps. Next derive a recursion relation in $N$, and take the continuum limit $N\to \infty$, see e.g. Refs. 4 & 5.

    • Evaluate a functional determinant, see e.g. Ref. 2 and this related Phys.SE post. Alternatively, use the Gelfand-Yaglom formula.

    • For $\omega\Delta t^E_{21}\ll 1$, one may use perturbative WKB methods.

    • If the Feynman propagator/kernel/amplitude $K(x_2,t^E_2;x_1,t^E_1)$ is known for the free particle, there is an ingenious trick to derive $K(x_2,t^E_2;x_1,t^E_1)$ for the harmonic oscillator, cf. Ref. 3.

  3. Once eq. (1) is a found, perhaps via handwaving arguments, there is a rigorous way to check it: Perform a single Gaussian integration over $x_2$ to check the path integral property $$K(x_3,t_3^E;x_1,t_1^E)~=~\int_{\mathbb{R}} \! dx_2~ K(x_3,t_3^E;x_2,t_2^E)~K(x_2,t_2^E;x_1,t_1^E), \tag{3}$$ which is the signature property for a sum over histories. Eq.(3) follows directly from eqs. (1)-(2), the Gaussian integration formula, and the addition formulas for $\coth$ & $\sinh$.

  4. In particular, if eq. (1) was originally only established for small times, $\omega\Delta t^E_{21}\ll 1$, then repeated application of eq. (3) can be used to establish eq. (1) for large times, in the very spirit of the path integration.

References:

  1. R.P. Feynman & A.R. Hibbs, Quantum Mechanics and Path Integrals, 1965; eqs. (3.59)-(3.60).

  2. J. Polchinski, String Theory Vol. 1, 1998, Appendix A.

  3. L. Moriconi, An Elementary Derivation of the Harmonic Oscillator Propagator, Am. J. Phys. 72 (2004) 1258, arXiv:physics/0402069. (Hat tip: OP.)

  4. S.M. Cohen, Path Integral for the Quantum Harmonic Oscillator Using Elementary Methods, Am. J. Phys. 66 (1998) 537.

  5. K. Hira, Eur. J. Phys. 34 (2013) 777.

Qmechanic
  • 201,751
  • Notes for later: Landau problem. 2D point particle in a magnetic field. Feynman & Hibbs, problem 3-10; Kleinert, subsection 2.23.3, p. 197. $\quad L~=~\frac{m}{2}(\dot{x}^2+\dot{y}^2)- U~\sim~\frac{1}{2}\begin{pmatrix}x & y \end{pmatrix}\begin{pmatrix} -m d_t^2 & Bd_t \cr -Bd_t& -m d_t^2 \end{pmatrix}\begin{pmatrix}x \cr y \end{pmatrix}$; $\quad -U~=~q(A_x \dot{x}+A_y \dot{y})~=~\frac{B}{2}(x \dot{y}-y \dot{x})$; $\quad B:=qB_z$; Position coordinates $x$ and $y$ do not commute, so it is inconsistent to require Dirichlet BC for both, cf. HUP. Let us instead impose periodic BC. – Qmechanic Nov 08 '18 at 12:21
  • Notes for later: $\quad \omega~=~\frac{2\pi n}{T}$; $\quad n\in \mathbb{N}$; $\quad \det\begin{pmatrix} m \omega^2-\lambda & iB\omega \cr -iB\omega& m \omega^2-\lambda \end{pmatrix}~=~0$; $\quad \lambda_{\pm}~=~ m \omega^2\pm B\omega$; $\quad \begin{pmatrix} -m d_t^2 -\lambda & Bd_t \cr -Bd_t& -m d_t^2 -\lambda \end{pmatrix}\begin{pmatrix}x \cr y \end{pmatrix}~=~\begin{pmatrix}0 \cr 0 \end{pmatrix}$; $\quad \begin{pmatrix}x_{\pm} \cr y_{\pm} \end{pmatrix}~=~\begin{pmatrix}\cos/\sin\omega t \cr \sin/\cos\omega t \end{pmatrix}$; Non-negative eigenvalues if $B$ is small enough. – Qmechanic Jun 07 '19 at 15:26
  • Notes for later: $\quad \det^{\prime}~=~\prod_{n\in\mathbb{N}} (m^2\omega^4-B^2\omega^2)~=~\left[\prod_{n\in\mathbb{N}}n\right]^2 \left[\prod_{n\in\mathbb{N}}m^2\omega_1^4\right]\left[\prod_{n\in\mathbb{N}} \left(1-\left(\frac{B}{m\omega_1 n}\right)^2\right)\right]=\frac{T^2}{m}{\rm sinc}\frac{BT}{2m}$ cf. zeta function regularization. – Qmechanic Jun 07 '19 at 16:17
  • Notes for later: Propagator in same position: $\quad K(x,t^E_2;x,t^E_1)~=~\sqrt{\frac{m\omega}{2\pi\hbar \sinh(\omega\Delta t^E_{21}) }}\exp\left{ -\frac{m\omega x^2}{\hbar}\tanh\frac{\omega\Delta t^E_{21}}{2}\right}$; Path integral: $\quad Z~=~\int_{\mathbb{R}}!dx~K(x,t^E_2;x,t^E_1)~=~\left(2\sinh\frac{\omega\Delta t^E_{21}}{2}\right)^{-1}~=~\sum_{n\in\mathbb{N}0}e^{-(n+1/2)\omega\Delta t^E{21}}$; – Qmechanic Jun 02 '20 at 11:24
  • Notes for later: What happens if we add a source term $Jx$ to the Lagrangian? Lagrangian $\quad L_M \sim -\frac{m}{2}x\left(\frac{d^2}{dt_M^2}+\omega^2\right)x+Jx.$ $\quad L_E \sim \frac{m}{2}x\left(-\frac{d^2}{dt_E^2}+\omega^2\right)x-Jx = \frac{m}{2}(x-G_E J)\left(-\frac{d^2}{dt_E^2}+\omega^2\right)(x-G_E J)-\frac{1}{2}JG_E J.$ Greens function $\quad G_M(t_M)=-{\rm sgn}(t_M)\frac{\sin(\omega t_M)}{2m\omega}.$ $\quad G_E(t_M)=-{\rm sgn}(t_E)\frac{\sinh(\omega t_E)}{2m\omega}.$ – Qmechanic Jan 28 '22 at 14:43
  • Notes for later: $\quad \langle x_2,t^E_2|x_1,t^E_1\rangle_J = \langle x_2-G_E(t^E_2)J,t^E_2|x_1-G_E(t^E_1)J,t^E_1\rangle_0~\exp\left(\frac{1}{2\hbar}\iint JG_EJ\right).$ Compare with M.S. Swanson, Section 3.2 eqs. (3.19)-(3.21); and Feynman & Hibbs, problem 3-11 eq. (3-66). – Qmechanic Jan 28 '22 at 19:58
2

The path integral in quantum mechanics may be defined as,

$$\int_{-\infty}^\infty \dots \int_{-\infty}^\infty \exp \left \{\frac{i}{\hbar}\Delta t \sum_i L \left(x_i,\frac{x_{i+1}-x_i}{\Delta t}, i \right) \right\} \, \mathrm dx_0 \dots \mathrm dx_N$$

where as the OP has noted, one 'slices' time into $N+1$ segments and the idea is that the propagator is given by the formal limit as $N \to \infty$. Based on this paper, it seems that convergence has been established by Fujikawa in the norm operator topology, in $\mathcal{B}(L^2(\mathbb R^d))$ providing the potential is smooth with at most quadratic growth (e.g. a harmonic oscillator).

This has been extended to show convergence remains, providing second space derivatives exist in $H^{d+1}(\mathbb R^d)$. These results show we can expect to indeed recover the original propagator in the continuum limit.

However, for any finite $N$, we cannot expect to do anything but approximate the propagator; we can of course carry out the integration finitely many times simply. In fact, this is what is originally done to notice the pattern that emerges, which enables taking the $N\to\infty$ limit.

JamalS
  • 19,254