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Write $\psi =\sqrt{\rho }{e}^{i\theta }$ away from zeros, and ${\mathbf{v}}_s=\nabla \theta$. Its Madelung equations give ${\rho }_t+\nabla \cdot (\rho {\mathbf{v}}_s)=0$ and ${\theta }_t+|{\mathbf{v}}_s{|}^2/2=1-\rho +{\nabla }^2\sqrt{\rho }/(2\sqrt{\rho })-\zeta D_n\rho$. For an incompressible normal fluid, $D_n\rho =-\nabla \cdot [\rho ({\mathbf{v}}_s-{\mathbf{v}}_n)]$. Taking a gradient proves the displayed equation when each space-time derivative of the slowly varying amplitudes is $O(ϵ)$: the quantum potential force is $O({ϵ}^3)$, while the bulk term is $O({ϵ}^2)$.

Writing the wavefunction in modulus and dimensionless quantum phase gives a probability continuity equation and a Hamilton-Jacobi equation with a quantum potential:The phase equations hold locally away from wavefunction nodes. They follow by equating real and imaginary parts of the Time-dependent Schrödinger equation.

A constant-amplitude plane wave has no density derivative and therefore no damping. The complete family iswith arbitrary constant phase $\chi$. The background in the question is the member $A=1$.

Use the exact amplitude–phase equations derived above with ${\mathbf{v}}_n=0$. To first order in the perturbation, $\rho =1+2ϵR$ and ${\mathbf{v}}_s=\mathbf{k}+ϵ\nabla \Theta$, soThese equations retain the quantum potential contribution because no long-wavelength truncation is assumed in this subpart. In particular damping contains $R_t$, not the advective derivative along the moving condensate: its reference is the stationary normal fluid.

For the proposed periodic perturbation, write $q=\mathbf{k}\cdot \mathbf{l}$ and $L=l^2$. The amplitude coefficients satisfyA nonzero perturbation therefore requires the plane-wave dispersion of a material-density-damped condensateor equivalently ${\sigma }^2+(2iq+\zeta l^2)\sigma +l^2+l^4/4-q^2=0$. At zero background flow the roots are $-\zeta l^2/2\pm \sqrt{{\zeta }^2{l}^4/4-l^2-l^4/4}$, whose real parts are negative for $l\ne 0$. As $\zeta \to 0$, $\sigma =-iq\pm i\sqrt{l^2+l^4/4}$ recovers the advected Bogoliubov spectrum with long-wavelength sound speed one in these units. This differs from question 1's time normalization, whose sound speed is $1/\sqrt{2}$.

Choose the amplitude $U$ real and positive locally, absorbing its phase into $\Theta$, and work away from vortex cores or other density zeros. Put $\rho =U^2$ and $\theta =\Theta /ϵ$. Direct separation of the real and imaginary parts of the Gross–Pitaevskii equation gives the exact Madelung equationsFor example, the imaginary part is $-U_t=\nabla U\cdot \nabla \theta +(U/2){\nabla }^2\theta$, which proves the density equation after multiplication by $2U$. The real damping coefficient affects the phase balance rather than adding a particle source.

Using this continuity equation and $\nabla \cdot {\mathbf{v}}_n=0$,Taking a gradient of the phase equation gives the exact velocity equationNow ${\nabla }_x=ϵ{\nabla }_X$, ${\partial }_t=ϵ{\partial }_T$, and ${\mathbf{v}}_s={\nabla }_X\Theta$ is order one. The inertial and chemical-potential gradients are $O(ϵ)$; the double-gradient bulk term is $O({ϵ}^2)$; and the quantum potential force contains three slow derivatives and is $O({ϵ}^3)$, provided $U$ stays bounded away from zero. Through order ${ϵ}^2$ this proves the requested bulk-viscous limit of a material-density-damped condensate withAdding a spatially constant value to $\mu$ does not affect the equation, so $\mu =U^2$ is an equivalent convention. The density and viscosity coefficients are in the equation's dimensionless units. The bulk viscosity term is the gradient of the divergence of the relative mass flux, not a shear-viscosity term. At density zeros the slow-amplitude expansion and the division by $U$ fail; there one must retain the original field equation.

For a positive uniform number density $n_{\ast }$ and constant superfluid velocity $v\hat{\mathbf{x}}$, current divergence vanishes. The stationary density equation therefore imposes $(\gamma -\Gamma n_{\ast })n_{\ast }=0$, giving $n_{\ast }=\gamma /\Gamma$. The quantum potential vanishes, so the phase equation sets $\mu =m{v}^2/2+V_0\gamma /\Gamma$. Thus the uniform-current pumped polariton condensate isDirect substitution verifies the equation: the kinetic term gives $m{v}^2/2$, the interaction gives $V_0\gamma /\Gamma$, and the gain-loss term is zero. A constant overall phase is arbitrary.

With the stated $\gamma ,\Gamma >0$, this nonzero plane wave exists for any real $v$ and real interaction coefficient $V_0$ when $\mu$ is determined by the solution. If a particular real chemical potential is prescribed instead, the existence condition isEquality gives the uniform condensate at rest. A nonzero density generally requires $\gamma /\Gamma >0$; a zero nonlinear-loss coefficient with positive pumping would not allow this nonzero stationary balance. The vacuum is a separate zero solution, unstable to small perturbations for positive pump.

Existence should not be confused with stability. For completeness, linearize in the comoving frame with $\psi ={\psi }_{\ast }[1+u+iw]$, and let ${ϵ}_k={\hslash }^2{k}^2/(2m)$. The linear real-amplitude and phase equations givesoFor $V_0\ge 0$ both growth rates have nonpositive real parts, including the neutral uniform-phase mode. For $V_0<0$, sufficiently small nonzero $k$ makes the constant term negative, so one growth rate is positive: the plane wave still exists but is unstable in an infinite system. Uniform drift adds only a Doppler phase to these growth rates in this model.

Define the physical velocity potential by $\psi =\sqrt{n}\exp (im\varphi /\hslash )$, so the superfluid velocity is $\mathbf{v}=\nabla \varphi$. The condensate number current isMultiplying the pumped Gross–Pitaevskii equation by ${\psi }^{\ast }$ and subtracting its complex conjugate gives the density equation. Equivalently, substituting the amplitude and phase and equating imaginary parts yieldsEquating real parts gives the other gain-loss Madelung equations relation,The last term is the quantum potential, retained wherever $n>0$. Taking a gradient gives the equivalent velocity equationin an irrotational region. In equilibrium $n_t=0$ and $\varphi (\mathbf{x},t)=\Phi (\mathbf{x})-\mu t/m$, soThis convention distinguishes the physical velocity potential from the dimensionless quantum phase $m\varphi /\hslash$; using the latter instead would change the divergence prefactor.

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 Schrödinger 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.

In de Broglie-Bohm theory, the system has both a wavefunction and an actual configuration of particle positions. The wavefunction obeys the Time-dependent Schrödinger equation; the configuration follows the guidance equation. For spinless particles without electromagnetic vector potentials, write $\Psi =R{e}^{iS/\hslash }$ using a dimensionful action $S$. ThenThe actual configuration selects a trajectory, while the guiding wave generally retains all branches. A measurement correlates system and apparatus configurations; the actual apparatus position lies in one recorded branch. Effective collapse follows when wave packets have separated sufficiently in configuration space, although the full wave continues its Schrödinger evolution.

Substituting the polar form into that equation yields the probability continuity equation and a modified Hamilton-Jacobi equation,The amplitude-dependent term is the quantum potential. Taking gradients gives a second-order force description, but initial velocities remain constrained by the guidance law. Nodes require care because the displayed quotient is singular there.

If the initial configuration distribution is $|\Psi {|}^2$, the same continuity equation preserves that distribution: quantum equilibrium equivariance. It then reproduces ordinary nonrelativistic quantum measurement statistics within the model's applicability. Equivariance preserves a distribution; it is not by itself a proof that every initial distribution equals $|\Psi {|}^2$.

For entangled particles the guiding wave lives on their joint configuration space, so one particle's velocity can depend on the distant actual configuration. This explicit nonlocality avoids the locality premise of a Bell inequality. In quantum equilibrium it does not permit controllable faster-than-light signalling. Spin extensions use a spinor wavefunction and its probability current; they need not assign a classical little spinning vector with simultaneous definite values for every spin component.

David Bohm developed this account in his two 1952 papers. The first sets out the dynamics and the second treats measurement and the relation to ordinary predictions.^[ref][ref] The theory supplies additional dynamical variables and a particular nonlocal account of outcomes, rather than deriving a unique interpretation from the standard wavefunction alone.

Bohmian mechanics retains definite particle positions and deterministic trajectories, which makes it resemble classical mechanics at one level. But the guiding dynamics depends on a wavefunction obeying the Schrödinger equation, with velocity schematically ${\dot{Q}}_i=(\hslash /m_i)\mathrm{Im}({\nabla }_i\psi /\psi )(Q)$. For many particles this wavefunction is defined on configuration space and can be entangled. The velocity of one particle can consequently depend on the configuration of distant particles. Classical-looking trajectories therefore do not entail classical locality or separability.

Writing $\psi =R{e}^{iS/\hslash }$ also yields a Hamilton-Jacobi-like equation with quantum potential $Q=-{\sum }_i{\hslash }^2{\nabla }_i^{2}R/(2m_{i}R)$. The quantum potential is not generally a sum of ordinary pair potentials in three-space; its configuration dependence carries the nonclassical correlations. A stationary real bound-state wavefunction can give a stationary particle configuration, unlike the orbit one might infer from a classical Coulomb picture. Measurement outcomes are contextual: the actual apparatus interaction helps determine which position record is produced.

With the quantum-equilibrium distribution $\rho =|\psi {|}^2$, the guidance and wave equations preserve that distribution and reproduce the usual quantum statistics. Determinism alone does not establish that distribution; equilibrium or typicality is an additional explanatory component. The theory also leaves philosophical choices about whether the wavefunction is a physical field, a law-like object or part of a broader ontology.

The significant departure is therefore not simply replacing chance by hidden classical motion. It is the entangled guiding structure, nonlocal dependence and contextual relation between measurements and properties. The retained particles make the account continuous with some classical intuitions, while its actual law and ontology depart substantially from classical many-body mechanics.^[ref][ref]

For a measurement specifically, the apparatus configuration occupies one region associated with a record. The guidance law evolves that actual configuration into one branch of the entangled system-apparatus wavefunction; disjoint unoccupied branches do not contain the actual pointer. In quantum equilibrium the distribution of those occupied outcome regions is the Born distribution. Thus the account supplies a definite record without requiring stochastic wavefunction collapse, while paying the costs of added configuration ontology, nonlocal guidance and a justified equilibrium measure.

Bohmian mechanics retains definite particle positions and deterministic trajectories, which makes it resemble classical mechanics at one level. But the guiding dynamics depends on a wavefunction obeying the Schrödinger equation, with velocity schematically ${\dot{Q}}_i=(\hslash /m_i)\mathrm{Im}({\nabla }_i\psi /\psi )(Q)$. For many particles this wavefunction is defined on configuration space and can be entangled. The velocity of one particle can consequently depend on the configuration of distant particles. Classical-looking trajectories therefore do not entail classical locality or separability.

Writing $\psi =R{e}^{iS/\hslash }$ also yields a Hamilton-Jacobi-like equation with quantum potential $Q=-{\sum }_i{\hslash }^2{\nabla }_i^{2}R/(2m_{i}R)$. The quantum potential is not generally a sum of ordinary pair potentials in three-space; its configuration dependence carries the nonclassical correlations. A stationary real bound-state wavefunction can give a stationary particle configuration, unlike the orbit one might infer from a classical Coulomb picture. Measurement outcomes are contextual: the actual apparatus interaction helps determine which position record is produced.

With the quantum-equilibrium distribution $\rho =|\psi {|}^2$, the guidance and wave equations preserve that distribution and reproduce the usual quantum statistics. Determinism alone does not establish that distribution; equilibrium or typicality is an additional explanatory component. The theory also leaves philosophical choices about whether the wavefunction is a physical field, a law-like object or part of a broader ontology.

The significant departure is therefore not simply replacing chance by hidden classical motion. It is the entangled guiding structure, nonlocal dependence and contextual relation between measurements and properties. The retained particles make the account continuous with some classical intuitions, while its actual law and ontology depart substantially from classical many-body mechanics.^[ref][ref]