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An instanton describes a finite-action trajectory in imaginary time that crosses between classical vacua. It accounts for quantum tunnelling effects exponentially small in $\hslash$, beyond an ordinary power series around one minimum. A symmetric double well makes the connection between the Euclidean path integral, fluctuation theory and the low-energy spectrum particularly explicit.

For a particle of mass $\mu$, use a Wick rotation $t=-i\tau$ in the kernel. The Euclidean action and its classical equation areThe latter is Newtonian motion in the inverted potential $-V$. Subtract the common minimum energy so $V(\pm a)=0$. A finite-action path approaching $\pm a$ at infinite Euclidean times has conserved Euclidean energy $\mu {\dot{q}}^2/2-V=0$. ThereforeThis gives both the classical trajectory by a quadrature and its tunnelling exponent. The plus solution is the instanton; reversal is the anti-instanton. A single crossing has action $S_I$, not $2S_I$, although a periodic closed path necessarily has equal numbers of crossings in both directions.

An explicit double well. Take $V(q)=\mu {\omega }^2(q^2-a^2{)}^2/(8a^2)$, whose local harmonic frequency at either minimum is $\omega$. Integrating the zero-energy equation gives the quartic double-well instantonThe arbitrary center ${\tau }_0$ is a collective coordinate. The tunnelling regime is $S_I/\hslash \gg 1$, so the crossing width ${\omega }^{-1}$ is short compared with the typical separation between rare crossings.

The exponential alone does not give a dimensionally complete amplitude. Expand $q=q_I+\eta$; the quadratic Euclidean action is $\mu \int \eta L_I\eta d\tau /2$, whereThe derivative ${\dot{q}}_I$ is a normalizable translation instanton translation zero mode. The factorization of this one-dimensional fluctuation operator shows it has no negative modes. In $z=\omega (\tau -{\tau }_0)/2$, its two discrete eigenfunctions are proportional to ${\mathrm{sech}}^{2}z$ and $\mathrm{sech}z\tanh z$, with eigenvalues zero and $3{\omega }^2/4$; the continuum begins at ${\omega }^2$. Treating its zero eigenvalue as an ordinary Gaussian integral would give a divergent determinant. Instead integrate over ${\tau }_0$: since $\mu \int {\dot{q}}_I^{2}d\tau =S_I$, the Jacobian supplies $\sqrt{S_I/(2\pi \hslash )}d{\tau }_0$. The nonzero fluctuations supply the determinant ratio. With common endpoint regularization,This is the instanton fluctuation prefactor, with $K$ and $\kappa$ having frequency units. The prime omits precisely the translation mode.

For this quartic potential the determinant can be evaluated explicitly. Set $z=\omega (\tau -{\tau }_0)/2$ and add a small positive spectral regulator $\epsilon$ to both operators. With $r=2\sqrt{1+\epsilon /{\omega }^2}$, the dimensionless operator is $-{\partial }_z^2+r^2-6{\mathrm{sech}}^{2}z$. Applying successively $-{\partial }_z+\tanh z$ and $-{\partial }_z+2\tanh z$ to the free decaying solution $e^{-rz}$, and normalizing at $z\to +\infty$, gives a growing coefficient at $z\to -\infty$ equal to $(r-1)(r-2)/[(r+1)(r+2)]$. The common Dirichlet functional determinant ratio has this coefficient. Hence the quartic double-well fluctuation determinant isThe frequency factor cannot be discarded when deleting the zero eigenvalue.

Summing rare crossings. Consider Euclidean duration $T$ much longer than a crossing width. In the dilute instanton gas, $n$ ordered centers have integration volume $T^n/n!$, and the amplitude per crossing is $\kappa$. Starting in the left well, directions alternate: an even number returns to the left well, and an odd number ends in the right well. If $E_{\mathrm{well}}=\hslash \omega /2+$ local anharmonic corrections, the leading kernels, with a common endpoint normalization $\mathcal{A}$, areThe even/odd sums must not be replaced by counting all possible crossing directions independently. Comparing their exponential components with the spectral representation givesFor the quartic example, $\Delta E=2\hslash \omega \sqrt{6S_I/(\pi \hslash )}{e}^{-S_I/\hslash }$ to leading semiclassical order. This is the double-well tunneling splitting: the even state is lower because the symmetric wavefunction has no node. The equivalent low-energy Hamiltonian in localized left/right states is $E_{\mathrm{well}}I-\hslash \kappa {\sigma }_x$. An initially localized state tunnels coherently, with probability ${\sin }^2(\Delta Et/(2\hslash ))$ of occupying the other well. Tunnelling does not imply irreversible decay in this closed symmetric system.

The instanton approach thus yields the same leading forbidden-region action as the WKB approximation, while identifying the zero-mode measure, fluctuation normalization and multiple tunnelling events systematically. Perturbation theory about either separate minimum cannot generate $e^{-S_I/\hslash }$. Multi-instanton interactions and higher-loop fluctuations correct the leading dilute-gas result; at finite temperature one instead uses periodic Euclidean boundary conditions, whose saddles may differ when the period is comparable to the crossing width. A false-vacuum bounce returning to the same metastable minimum has a negative fluctuation mode and describes a decay rate. It should not be confused with the stable-vacuum connecting instanton above, which has a translation zero mode but no negative mode.

The time-independent Schrodinger equation gives the even interior wavefunction $\psi =A\cos (\alpha x)$, with ${\alpha }^2=2m(U+E)/{\hslash }^2$. Outside, normalizability of a bound state requires $\psi =C{e}^{-\beta (|x|-l)}$, with ${\beta }^2=-2mE/{\hslash }^2$.

Continuity of the wavefunction and its derivative at $x=l$ gives $C=A\cos (\alpha l)$ and $-\beta C=-\alpha A\sin (\alpha l)$. A nonzero solution cannot have $\cos (\alpha l)=0$, so division givesAt the formally allowed endpoint $E=-U$, the even interior solution is constant; derivative matching would force the exterior amplitude to vanish, leaving the zero solution. Nontrivial bound states therefore have $-U<E<0$. The matching formula is the even-state condition for a finite square well.

In Bohmian mechanics the particle has a definite position $\mathbf{X}(t)$ at every time. Its wavefunction obeys the usual autonomous wave equation, while its actual position follows the guidance equationHere $\mathbf{j}$ is the probability current. The Born rule is the quantum-equilibrium choice of initial position distribution $\rho =|\psi {|}^2$; quantum equilibrium equivariance ensures that this distribution persists because it obeys the same probability continuity equation as the wave amplitude. The guidance equation fixes the initial velocity as well as subsequent velocities: the second-order equation below does not permit an independent arbitrary initial velocity.

Define the quantum potentialTaking the gradient of the real Madelung equations givesOn any smooth phase patch $\nabla \times \mathbf{v}=0$, so $\nabla (|\mathbf{v}{|}^2/2)=(\mathbf{v}\cdot \nabla )\mathbf{v}$. Along the actual path, differentiation is the material derivative $D/Dt={\partial }_t+\mathbf{v}\cdot \nabla$. ThereforeThis is the Bohmian mechanics Newton form: the classical force is supplemented by the amplitude-dependent quantum potential. Neither division by $R$ nor a smooth phase is justified at a wavefunction node, so the derivation applies on nonzero-amplitude regions. A nonzero circulation around a node is compatible with the locally curl-free guidance equation.

Use the dimensionless quantum phase $S$, so that $\psi =R{e}^{iS}$ with $R\ge 0$. Work locally away from wavefunction nodes, where $R$ and $S$ are differentiable. Differentiating the wavefunction for substitution in the Time-dependent Schrodinger equation gives:Cancel $e^{iS}$ in the time-dependent equation and equate real and imaginary parts. The resulting Madelung equations areThe second equation is a Hamilton-Jacobi equation for the action $\hslash S$, with an additional quantum potential. To see the meaning of the first, multiply it by $2R$ and set $\rho =R^2$. It becomes the probability continuity equationThus the probability density is transported by the velocity field $\hslash \nabla S/m$. The Madelung equations are a local rewriting of the linear wave equation; their apparent nonlinearity comes from expressing a complex wavefunction in modulus and phase variables.