Quantum Mechanics Course

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Part IX, Chapter 1 | Page 2 of 3

Exact Solutions and Semiclassical Methods

Free particle, harmonic oscillator, and the WKB approximation from path integrals

Free Particle Propagator

The simplest exactly solvable path integral is the free particle ($V = 0$). The discretized propagator becomes a product of Gaussian integrals. Starting from the time-sliced expression with $N$ intervals of width $\varepsilon$:

$$K_{\text{free}} = \left(\frac{m}{2\pi i\hbar\varepsilon}\right)^{N/2}\!\int\!\prod_{j=1}^{N-1}dx_j\;\exp\!\left[\frac{im}{2\hbar\varepsilon}\sum_{k=0}^{N-1}(x_{k+1}-x_k)^2\right]$$

Each integration is Gaussian. We evaluate them sequentially, starting from $x_1$. The integral over $x_1$ couples the first two quadratic terms:

$$\int dx_1 \exp\!\left[\frac{im}{2\hbar\varepsilon}\!\left((x_1 - x_0)^2 + (x_2 - x_1)^2\right)\right] = \sqrt{\frac{\pi i\hbar\varepsilon}{m}}\cdot\frac{1}{\sqrt{2}}\exp\!\left[\frac{im(x_2-x_0)^2}{2\hbar\cdot 2\varepsilon}\right]$$

After $j$ integrations, the pattern becomes $\propto (j+1)^{-1/2}\exp\!\left[\frac{im(x_{j+1}-x_0)^2}{2\hbar(j+1)\varepsilon}\right]$.

After completing all $N-1$ integrations and collecting prefactors (a telescoping product gives$N^{-1/2}$), with $T = N\varepsilon = t_b - t_a$:

$$\boxed{K_{\text{free}}(x_b, t_b;\, x_a, t_a) = \sqrt{\frac{m}{2\pi i\hbar T}}\;\exp\!\left[\frac{im(x_b - x_a)^2}{2\hbar T}\right]}$$

where $T = t_b - t_a$. The exponent is precisely $iS_{\text{cl}}/\hbar$ with $S_{\text{cl}} = m(x_b-x_a)^2/2T$.

Verification: Plane Wave Propagation

We can verify this result by propagating a plane wave $\psi(x,0) = e^{ipx/\hbar}$. Inserting into the propagation formula:

$$\psi(x_b, T) = \int K_{\text{free}}(x_b, T;\, x_a, 0)\, e^{ipx_a/\hbar}\, dx_a = \exp\!\left[\frac{i}{\hbar}\!\left(px_b - \frac{p^2}{2m}T\right)\right]$$

This is exactly $e^{i(px_b - Et)/\hbar}$ with $E = p^2/2m$ — the correct free-particle energy-momentum relation.

Harmonic Oscillator Propagator

For $V = \frac{1}{2}m\omega^2 x^2$, the action is quadratic in $x(t)$, so the path integral is again exactly Gaussian. We decompose any path into the classical path plus fluctuations: $x(t) = x_{\text{cl}}(t) + y(t)$ where $y(t_a) = y(t_b) = 0$.

Since the action is quadratic, it separates cleanly:

$$S[x_{\text{cl}} + y] = S[x_{\text{cl}}] + \underbrace{\frac{\delta S}{\delta x}\bigg|_{x_{\text{cl}}}}_{= 0}\!\cdot y + \frac{1}{2}\frac{\delta^2 S}{\delta x^2}\bigg|_{x_{\text{cl}}}\!\cdot y^2$$

The linear term vanishes because $x_{\text{cl}}$ satisfies the Euler-Lagrange equation. For quadratic potentials, there are no higher-order terms.

Classical Action

The classical path for the harmonic oscillator with boundary conditions$x_{\text{cl}}(t_a) = x_a$ and $x_{\text{cl}}(t_b) = x_b$ is:

$$x_{\text{cl}}(t) = \frac{x_a\sin\omega(t_b - t) + x_b\sin\omega(t - t_a)}{\sin\omega T}$$

Evaluating the action $S_{\text{cl}} = \int_{t_a}^{t_b}\frac{m}{2}(\dot{x}^2 - \omega^2 x^2)\,dt$ along this path:

$$S_{\text{cl}} = \frac{m\omega}{2\sin\omega T}\left[(x_a^2 + x_b^2)\cos\omega T - 2x_a x_b\right]$$

Fluctuation Determinant

The fluctuation integral over $y(t)$ with $y(t_a) = y(t_b) = 0$ gives the Van Vleck determinant. For the harmonic oscillator, the fluctuation operator is$\hat{A} = -m(\partial_t^2 + \omega^2)$, and its functional determinant (relative to the free particle) can be evaluated via the Gel'fand-Yaglom method:

$$\frac{\det(-\partial_t^2 - \omega^2)}{\det(-\partial_t^2)} = \frac{\sin\omega T}{\omega T}$$

This ratio of determinants is computed from the solution to $(-\partial_t^2 - \omega^2)f = 0$ with $f(0) = 0$, $f'(0) = 1$: namely $f(T) = \sin\omega T/\omega$.

The Mehler Kernel

Combining the classical action with the fluctuation determinant yields the complete harmonic oscillator propagator (the Mehler kernel):

$$\boxed{K_{\text{HO}}(x_b, t_b;\, x_a, t_a) = \sqrt{\frac{m\omega}{2\pi i\hbar\sin\omega T}}\;\exp\!\left[\frac{im\omega}{2\hbar\sin\omega T}\!\left((x_a^2 + x_b^2)\cos\omega T - 2x_a x_b\right)\right]}$$

Setting $\omega \to 0$ recovers the free particle propagator. The singularity at $\omega T = n\pi$ corresponds to classical focal points (caustics).

Semiclassical (WKB) Approximation

For a general potential, the path integral cannot be evaluated exactly. The semiclassical approximation (equivalent to WKB) retains the dominant contribution from the classical path plus Gaussian fluctuations around it.

Write $x(t) = x_{\text{cl}}(t) + y(t)$ and expand the action to second order:

$$S[x_{\text{cl}} + y] \approx S_{\text{cl}} + \frac{1}{2}\int_{t_a}^{t_b}\!\!\int_{t_a}^{t_b} y(t)\left[-m\delta(t-t')\!\left(\frac{d^2}{dt^2} + \omega_{\text{cl}}^2(t)\right)\right]y(t')\,dt\,dt'$$

where $\omega_{\text{cl}}^2(t) = V''(x_{\text{cl}}(t))/m$ is the local curvature of the potential along the classical path.

Performing the Gaussian functional integral over $y(t)$ gives the Van Vleck-Morette formula:

$$\boxed{K_{\text{sc}}(x_b, t_b;\, x_a, t_a) = \frac{1}{\sqrt{2\pi i\hbar}}\left|\frac{\partial^2 S_{\text{cl}}}{\partial x_b\,\partial x_a}\right|^{1/2}\!\exp\!\left[\frac{i}{\hbar}S_{\text{cl}} - \frac{i\pi}{2}\nu\right]}$$

where $\nu$ is the Morse index (number of conjugate points along the classical trajectory). The prefactor$|\partial^2 S_{\text{cl}}/\partial x_b\partial x_a|^{1/2}$ is the Van Vleck determinant.

Stationary Phase Method

The semiclassical approximation is a special case of the stationary phase method. For an integral of the form $I = \int f(x) e^{i\lambda g(x)}\,dx$ with $\lambda \gg 1$:

$$I \approx \sum_{\bar{x}:\, g'(\bar{x})=0} f(\bar{x})\sqrt{\frac{2\pi}{\lambda|g''(\bar{x})|}}\;\exp\!\left[i\lambda g(\bar{x}) + \frac{i\pi}{4}\text{sgn}\,g''(\bar{x})\right]$$

Applied to the path integral with $\lambda = 1/\hbar$, the stationary points$\delta S/\delta x = 0$ are classical paths, and the second variation determines the fluctuation prefactor.

When multiple classical paths exist between the same endpoints (as in tunneling problems or near caustics), the semiclassical propagator sums over all such paths:

$$K_{\text{sc}} = \sum_{\text{classical paths}} A_n\, e^{iS_n/\hbar - i\pi\nu_n/2}$$

The interference between different classical paths produces phenomena such as supernumerary rainbows and orbit resonances in atomic physics.

Gaussian Path Integrals: General Theory

Any quadratic action $S = \frac{1}{2}\int x\, \hat{A}\, x$ with differential operator $\hat{A}$ yields an exact Gaussian integral:

$$\int\mathcal{D}x\;\exp\!\left[\frac{i}{2\hbar}\int x\,\hat{A}\,x\,dt\right] = \left(\det\hat{A}\right)^{-1/2}$$

This is the functional analog of $\int \frac{dx}{\sqrt{2\pi}}\,e^{-ax^2/2} = a^{-1/2}$. Computing functional determinants is the key technical challenge.

Practical methods for evaluating $\det\hat{A}$ include:

  • Gel'fand-Yaglom theorem: Relates the determinant to a simple initial-value ODE problem
  • Zeta-function regularization: $\ln\det\hat{A} = -\zeta_{\hat{A}}'(0)$ where $\zeta_{\hat{A}}(s) = \sum_n \lambda_n^{-s}$
  • Heat kernel expansion: Asymptotic expansion of $\text{Tr}(e^{-t\hat{A}})$ for small $t$
  • Ratio of determinants: Compare with a known reference operator to cancel divergences

Charged Particle in a Uniform Field

Another exactly solvable case: a particle of charge $q$ in a uniform electric field $\mathcal{E}$, with Lagrangian $L = \frac{m\dot{x}^2}{2} + q\mathcal{E}x$. The action is quadratic in $x(t)$, so the path integral is Gaussian:

$$K_{\mathcal{E}} = \sqrt{\frac{m}{2\pi i\hbar T}}\;\exp\!\left[\frac{i}{\hbar}\left(\frac{m(x_b-x_a)^2}{2T} + \frac{q\mathcal{E}(x_a+x_b)T}{2} + \frac{q^2\mathcal{E}^2 T^3}{24m}\right)\right]$$

The prefactor is unchanged from the free particle (the fluctuation determinant depends only on the quadratic part of the potential). The phase contains the classical action for uniformly accelerated motion: the particle follows a parabolic trajectory, with quantum fluctuations around it.

General Properties of Propagators

Regardless of the potential, all propagators satisfy fundamental consistency conditions:

  • Initial condition: $\lim_{T\to 0^+}K(x_b,T;\,x_a,0) = \delta(x_b - x_a)$
  • Composition: $K(x_c,t_c;\,x_a,t_a) = \int K(x_c,t_c;\,x_b,t_b)\,K(x_b,t_b;\,x_a,t_a)\,dx_b$
  • Complex conjugate: $K^*(x_b,t_b;\,x_a,t_a) = K(x_a,t_a;\,x_b,t_b)$ (time reversal)
  • Spectral representation: $K = \sum_n \psi_n(x_b)\psi_n^*(x_a)\,e^{-iE_n T/\hbar}$

The spectral representation connects the path integral to the energy eigenvalue problem. In the limit$T \to \infty$ (with Wick rotation), only the ground state survives: $K_E \sim \psi_0(x_b)\psi_0^*(x_a)\,e^{-E_0\tau/\hbar}$.

Key Insight: For any system with a quadratic Lagrangian (free particle, harmonic oscillator, uniform fields), the path integral is exactly Gaussian. The propagator takes the universal form $K = F(T)\,e^{iS_{\text{cl}}/\hbar}$ where $F(T)$comes from the fluctuation determinant and $S_{\text{cl}}$ is the classical action. The semiclassical approximation extends this to general potentials by treating the neighborhood of the classical path as locally quadratic.

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