Interpretations of Quantum Mechanics · Chapter 06
What changes when particles have positions in addition to a wavefunction?
Starting point: Chapters 1–5; familiarity with density operators and linear algebra.
Anchor example: Probability current, a travelling wave, and an apparatus with separated packets.
Method: Calculate a concrete experiment, identify the assumptions, then compare what different accounts explain.
Watch for: Definite positions do not imply pre-existing values for every measured operator.
This lesson includes explanations, worked examples, and problems with solutions. Advanced results whose proofs are outside the chapter are identified explicitly.
What changes when particles have positions in addition to a wavefunction?
Bohmian mechanics supplements the wavefunction with an actual configuration Q(t). For N nonrelativistic spinless particles without vector potentials, the wavefunction obeys the usual Schrödinger equation on configuration space. The positions obey a guiding equation. Writing the law explicitly prevents the vague image of tiny classical balls following Newtonian trajectories.
\[\dot Q_k=\left.\frac{\hbar}{m_k}\operatorname{Im}\frac{\nabla_k\Psi}{\Psi}\right|_{q=Q},\qquad j_k=\frac{\hbar}{m_k}\operatorname{Im}(\Psi^*\nabla_k\Psi).\]Away from nodes, the velocity is jₖ/|Ψ|². For a local polar form Ψ=R exp(iS/ℏ), it is ∇ₖS/mₖ. The actual configuration and the wavefunction jointly determine the motion. Spin and electromagnetic potentials require the appropriate generalized current; the displayed formula is deliberately restricted.
For a plane wave on a periodic line, Ψ∝exp(ikx), the velocity is ℏk/m. For a nondegenerate stationary state with a real spatial wavefunction, the spatial phase gradient is zero and the velocity vanishes away from nodes. This may surprise classical intuition, but it follows directly from the stated theory. One should assess the model by its full measurement predictions rather than replacing its law with a classical guess.
Worked check 6.1. For Ψ=R(x)e^{ikx} with positive real R, find v.
The logarithmic derivative is R′/R+ik. Only k contributes to its imaginary part, giving v=ℏk/m.
Schrödinger evolution implies the continuity equation
\[\partial_t|\Psi|^2+\sum_k\nabla_k\cdot j_k=0.\]An ensemble of configurations transported by the guiding velocity satisfies \(\partial_t\rho+\sum_k\nabla_k\cdot(\rho v_k)=0\). Because jₖ=|Ψ|²vₖ, the density |Ψ|² obeys the same transport equation. Subject to the usual regularity and boundary conditions, if ρ=|Ψ|² initially, this equality is preserved. This property is called equivariance.
Equivariance does not say that every arbitrary initial distribution magically becomes |Ψ|². Quantum equilibrium is an additional statistical condition; typicality accounts explain its role by considering suitable measures on configurations, while relaxation accounts investigate particular nonequilibrium dynamics. These arguments should not be conflated with the continuity-equation identity.
In equilibrium, ensembles reproduce the familiar position probabilities and, with a suitable analysis of measurement devices, the usual quantum statistics. The Born density has a precise dynamical role here: it is preserved by the combined wave and configuration dynamics. Its interpretation and justification are still distinct questions from preservation.
Worked check 6.2. Does equivariance show that ρ=2|Ψ|² is a distinct equilibrium density?
No. It is not normalized when Ψ is normalized. Equivariance concerns evolution of the normalized |Ψ|² density; arbitrary alternatives require their own initial conditions.
Let an apparatus interaction produce two macroscopically disjoint packets in configuration space, Ψ=αΨ₀+βΨ₁. The actual configuration lies in one packet's support. When the other packet has negligible support near that configuration, the guiding equation is effectively governed by the occupied packet. This is effective collapse. It is not a literal deletion of the other wave component.
If the packets later overlap coherently, the neglected component can affect motion again. The effective wavefunction is therefore justified by dynamical separation, not by an irrevocable declaration that unused terms are nonexistent. This will matter for hypothetical measurements on entire laboratories.
For entangled Ψ, the velocity of one particle generally depends on the whole configuration, including distant coordinates. Bohmian mechanics is nonlocal in this sense. In equilibrium it nevertheless reproduces operational no-signalling. A relativistic extension is a further task; the nonrelativistic equation here is not already a Lorentz-covariant completion.
Finally, an operator measurement can depend on the full experimental arrangement. The theory does not assign every Hermitian operator a context-independent intrinsic value. Definite positions therefore do not evade the contextuality theorem by pretending its impossible value table exists.
Worked check 6.3. Why can an empty packet matter after recombination?
The velocity depends on the full wavefunction and its gradient at the actual configuration. Overlapping packets contribute interference terms there even if only one was occupied while they were separated.
These exercises draw on the whole chapter. Attempt each question before opening a hint, explain your reasoning, then compare it with the worked solution.
For a real stationary spatial wavefunction, calculate velocity away from nodes.
The ratio of gradient to wavefunction is real.
Its imaginary part vanishes, so the velocity is zero. The global time-dependent phase does not change the spatial derivative.
Why exclude nodes when writing ∇Ψ/Ψ?
Inspect the denominator.
The expression is undefined at Ψ=0. Trajectory existence and behavior near nodes require additional mathematical care; the local formula assumes nonzero density.
For a ring of length L with k=2πn/L, find the speed.
Use the plane-wave guiding law.
The signed velocity is 2πℏn/(mL), with magnitude its absolute value.
Substitute ρ=|Ψ|² into the configuration continuity equation.
Use ρv=j.
It becomes exactly the Schrödinger continuity equation. This establishes preservation of the equilibrium distribution under appropriate regularity conditions.
Does equivariance prove relaxation from any initial distribution?
Distinguish invariance from attraction.
No. Invariance says an initially matching density remains matching. Relaxation of other densities is a separate dynamical claim and is not universally guaranteed.
Why does an equilibrium ensemble reproduce pointer frequencies?
Partition configuration space into disjoint record regions.
Their probabilities are integrals of |Ψ|² over those regions. For normalized separated record packets, these give the corresponding squared amplitudes.
Is effective collapse a modification of Schrödinger evolution?
Track the full wavefunction.
No. The full state continues evolving unitarily. One packet suffices approximately for guidance while supports remain separated near the actual configuration.
Does Bell’s theorem exclude this theory because it has hidden variables?
Check the factorization assumption.
No. The guiding dynamics is generally nonlocal for entangled states. It does not satisfy the Bell-local assumption excluded by the inequality.
Why does a definite position not imply a predetermined value of every observable?
An apparatus may couple and evolve before a pointer is read.
The recorded value depends on wavefunction, configuration, and measurement arrangement. It need not reveal a context-independent property associated with the operator.
State a limitation of the equations in this chapter.
Read the physical scope.
They describe nonrelativistic spinless particles without vector potentials. Spin, gauge fields, particle creation, and relativistic structure need additional formulation.