Bohm's Equation of Motion

• Bohm's Equation of Motion is a deterministic formulation in the de Broglie–Bohm framework, combining the Schrödinger equation with particle trajectories guided by a quantum potential.
• The Newtonian reformulation employs a second-order guidance law that mirrors classical dynamics, ensuring consistency with standard quantum mechanics when a specific initial velocity is enforced.
• Relaxing the velocity constraint yields unphysical behavior, as numerical and analytical studies show trajectories diverging from the prescribed probability distribution.

Bohm’s Equation of Motion describes the deterministic dynamics of quantum systems within the de Broglie–Bohm framework, also known as Bohmian mechanics or the pilot-wave theory. This formalism postulates that a quantum system is characterized by both a wave function, evolving according to the Schrödinger equation, and particle trajectories guided by the phase of the wave function. Bohm’s key innovation was reformulating de Broglie’s original first-order guidance law as a second-order Newton-type equation of motion incorporating a quantum potential derived from the wave function itself. The theory is dynamically equivalent to standard quantum mechanics when a specific initial velocity constraint is imposed, but relaxing this constraint leads to physically untenable dynamics, as established by both numerical and analytical arguments (Goldstein et al., 2013).

1. Derivation from the Schrödinger Equation

Starting from the non-relativistic, single-particle Schrödinger equation,

$$i\hbar\,\partial_t\Psi = \Bigl(-\frac{\hbar^2}{2m}\nabla^2 + V(x)\Bigr)\,\Psi,$$

the wave function is cast in polar form: $$\Psi(x,t)=R(x,t)\,\exp\bigl\{i\,S(x,t)/\hbar\bigr\},$$ with $R \geq 0$ and $S$ real. Inserting this into the Schrödinger equation and separating real and imaginary parts yields:

• The continuity equation:

$$\partial_t (R^2) + \nabla\cdot\bigl(R^2 \frac{\nabla S}{m} \bigr) = 0,$$

expressing conservation of probability.

• The modified Hamilton–Jacobi equation:

$$\partial_t S + \frac{(\nabla S)^2}{2m} + V + Q = 0,$$

where the quantum potential is defined as

$$Q(x,t) = -\frac{\hbar^2}{2m}\frac{\nabla^2 R(x,t)}{R(x,t)}.$$

The trajectory $x(t)$ of the particle is postulated to follow the guidance equation: $\frac{dx}{dt} = \frac{1}{m} \nabla S(x(t),t).$

2. Newtonian Formulation with the Quantum Potential

Differentiating the guidance equation in time and substituting using the Hamilton–Jacobi relation, Bohm derived a second-order equation reminiscent of classical Newtonian dynamics: $$m\frac{d^2x}{dt^2} = -\nabla\bigl( V(x) + Q(x,t) \bigr).$$ This “quantum Newton law” incorporates both the classical potential $$\Psi(x,t)=R(x,t)\,\exp\bigl\{i\,S(x,t)/\hbar\bigr\},$$0 and the quantum potential $$\Psi(x,t)=R(x,t)\,\exp\bigl\{i\,S(x,t)/\hbar\bigr\},$$1, the latter encoding all non-classical effects and inherently depending on the wave function. The explicit dependence of $$\Psi(x,t)=R(x,t)\,\exp\bigl\{i\,S(x,t)/\hbar\bigr\},$$2 on the wave function amplitude leads to nonlocal and context-dependent behavior, a central feature of Bohmian mechanics.

3. Velocity Constraint and Dynamical Equivalence

Bohm’s second-order formulation requires an initial velocity constraint: $$\Psi(x,t)=R(x,t)\,\exp\bigl\{i\,S(x,t)/\hbar\bigr\},$$3 Imposing this ensures that $$\Psi(x,t)=R(x,t)\,\exp\bigl\{i\,S(x,t)/\hbar\bigr\},$$4 for all $$\Psi(x,t)=R(x,t)\,\exp\bigl\{i\,S(x,t)/\hbar\bigr\},$$5, so the trajectory remains on the surface specified by the phase gradient. Thus, Bohm’s quantum Newton law is dynamically equivalent to de Broglie’s first-order guiding law if and only if this initial constraint is enforced. Both formulations then yield identical ensembles of Bohmian trajectories and preserve the Born rule ($$\Psi(x,t)=R(x,t)\,\exp\bigl\{i\,S(x,t)/\hbar\bigr\},$$6 equivariance) (Goldstein et al., 2013).

4. Instability and Unphysical Behavior Without the Constraint

If the initial velocity constraint is relaxed, the resulting phase space dynamics display generic instability. Both numerical simulations—for example, in the work of Colin & Valentini—and analytic calculations for explicit wave functions (including free-particle Gaussians, coherent states in a harmonic oscillator, and bound eigenstates in central potentials) demonstrate that particle trajectories with momenta not equal to $$\Psi(x,t)=R(x,t)\,\exp\bigl\{i\,S(x,t)/\hbar\bigr\},$$7 escape irreversibly from the support of $$\Psi(x,t)=R(x,t)\,\exp\bigl\{i\,S(x,t)/\hbar\bigr\},$$8. Even in the presence of confining potentials, the total effective potential $$\Psi(x,t)=R(x,t)\,\exp\bigl\{i\,S(x,t)/\hbar\bigr\},$$9 is typically bounded above by the eigenenergy, so any excess kinetic energy supplied by violating the initial velocity constraint causes the particle to escape to infinity. Thus, the Bohmian “extended nonequilibrium” (momenta not locked to $R \geq 0$0) leads to unphysical outcomes with no relaxation back to quantum equilibrium, contradicting quantum empirical observations (Goldstein et al., 2013, Colin et al., 2013).

5. Illustrative Examples

Several instructive cases illustrate the instability of the unconstrained Bohmian dynamics:

• Free Gaussian wave packet: For any initial velocity $R \geq 0$1, the trajectory outruns the spreading packet, escaping to infinity. Only $R \geq 0$2 (the phase-gradient initial condition) yields a trajectory that tracks the packet width.
• Coherent state in a harmonic oscillator: The unconstrained case leads to linear drift and ultimate escape, whereas the guiding law trajectory oscillates bound to the packet center.
• Bound energy eigenstates in a central potential: Except on the constraint surface ($R \geq 0$3), the particle can run off to spatial infinity; positive values of energy-related constants eliminate classical turning points.

These results confirm that the quantum potential, by itself, does not guarantee bounded behavior nor maintains the probability distribution prescribed by the wave function unless supplemented by the strict velocity constraint (Goldstein et al., 2013).

6. Physical Interpretation and Theoretical Significance

Bohm’s guiding law, with the velocity field $R \geq 0$4, uniquely ensures the equivariance of the $R \geq 0$5 probability distribution under the Schrödinger flow and underpins empirical quantum mechanical predictions. The quantum potential encapsulates non-classical, nonlocal effects and is essential to the reproduction of quantum phenomena in a deterministic, trajectory-based framework. The second-order (Newtonian) quantum dynamics are only physically legitimate when phase-space configurations are restricted to those compatible with the first-order guidance law. Relaxing this requirement leads to theoretical and empirical inconsistency, rendering unconstrained Bohmian “quantum potential dynamics” untenable (Goldstein et al., 2013, Colin et al., 2013).

7. Applications and Further Developments

Bohm’s equation of motion provides a theoretical and computational tool for simulating wave function evolution, quantum hydrodynamics, and the analysis of quantum-classical correspondence—always under the constraint of guiding velocities. Further research has extended the Bohmian formalism to novel contexts, including entropic dynamics (where the causal or Bohmian limit emerges in the drift-dominated regime) and wave function reconstruction problems where analytic phase retrieval relies on the coupling between amplitude and phase via Bohm’s equations (Bartolomeo et al., 2015, Brasil et al., 2020). The guiding equation remains central for efforts to “engineer” quantum states or potentials and analyze the correspondence between macroscopic and subquantum models, always resting on the necessity of the phase-gradient velocity constraint enforced by the original de Broglie–Bohm theory.