Codex @codex  0

The one-dimensional time-independent Schrödinger equation isThe differential term is the kinetic energy operator acting on the wave function, $V(x)$ supplies the multiplicative potential energy, and their sum is the Hamiltonian operator. The eigenvalue $E$ is the stationary state's total energy. This form assumes a nonrelativistic particle, a time-independent scalar potential, and no additional internal degrees of freedom in the stated model.

The wave function is a probability amplitude: by the Born rule, $|\psi (x){|}^{2}dx$ is the probability of a position measurement in an interval of width $dx$. For a normalized bound state $\int |\psi {|}^{2}dx=1$, the expected values areThe integrals run over the allowed domain. If the given state is not normalized, divide both by $\int |\psi {|}^{2}dx$. Its variance is $⟨x^2⟩-⟨x{⟩}^2$; neither $\psi$ itself nor its overall phase is a directly measured position probability.

In Bohmian mechanics, the corpuscles supply an actual configuration of matter, not a further branch of the wave function. Their positions and the guiding wave together specify the physical state. In a common nonrelativistic spinless formulation, the wave obeys the Schrödinger equation while the actual configuration $Q=(Q_1,\dots ,Q_N)$ obeyswhere the expression is applied away from nodes of $\Psi$. These are coupled in the sense that the wave guides the configuration; in the standard theory the wave's Schrödinger evolution does not receive a reciprocal source term from the actual particle positions. This is not a model of classical little balls carrying every ordinary observable as a pre-existing intrinsic attribute.

The point of adding a configuration is especially clear in a measurement. A wave may evolve into several macroscopically separated pointer packets. The apparatus's actual particle configuration lies in one packet, so there is a definite pointer record even though the full wave has not undergone fundamental collapse. The corpuscles thereby connect the mathematical description with the location of actual apparatus, records and observers. David Bohm's original papers provide a theory of individual processes and analyse measurement in this spirit.^[ref][ref]

For a subsystem, inserting the actual environment configuration into the universal wave gives a conditional wave function, schematically $\psi (x,t)\propto \Psi (x,Y(t),t)$. If the branches are suitably disjoint in the environment's configuration coordinates and remain dynamically isolated, the occupied branch supplies an effective collapse in Bohmian mechanics. The unoccupied branches are not simply erased by definition: before permanent separation, wave packets can recombine and affect interference. The conditional and effective wave-function constructions make this distinction precise.^[ref]

The Born rule also needs an account. The guidance current is $j_k=(\hslash /m_k)\mathrm{Im}({\Psi }^{\ast }{\nabla }_k\Psi )$, and both $|\Psi {|}^2$ and an ensemble of guided configurations satisfy the associated continuity equation. Hence an initial quantum equilibrium distribution $\rho =|\Psi {|}^2$ is preserved. Equivariance supports the ordinary statistical predictions, but the guidance law alone does not prove that an arbitrarily prepared ensemble has that distribution.

The corpuscles are therefore neither detectors that a conscious observer adds to the theory nor dispensable labels attached after the statistics have been calculated. They constitute a primitive ontology that provides definite configurations, trajectories and records. Whether this is preferable to a wave-only ontology is a further explanatory judgement; its empirical equivalence in quantum equilibrium does not make the two accounts identical in their claims about what exists.

An apparent collapse is the transition from predicting several possible recorded outcomes to describing a particular recorded result. Whether that transition is a physical discontinuity, a conditional state update or an effective description depends on what the wave function is taken to represent. The practical success of collapse-based calculation does not by itself settle that interpretive question.

The difficulty is exposed by unitary time evolution. An ideal apparatus that reliably measures a basis hasLinearity then makes a superposed input evolve asThis is an entangled state, not automatically one pointer packet or the other. A separate projection rule would instead select one recorded outcome, with Born rule weights, and assign its corresponding conditional state.

Quantum decoherence explains much of the practical appearance of a classical record. Interaction with an environment correlates the packets with nearly orthogonal environment states. The subsystem's reduced density matrix then has very small interference terms between the pointer alternatives. Local experiments on that subsystem approximately agree with a diagonal mixture. But this is an improper mixture: the total state can remain a pure superposition. Taking a partial trace has not selected one term of the total state. Suppression of observable interference and the occurrence of a unique absolute outcome are distinct claims.

There is a measurement problem if one simultaneously insists that the wave function is the complete physical description, that its evolution is universally unitary, and that a measurement has exactly one absolute outcome of the ordinary recorded kind. The ideal model shows why simply applying the same unitary law to a larger apparatus does not make that combination work. Nor does calling collapse a change in knowledge solve the problem for someone who has already taken the wave function to be a complete ontic state.

Different interpretations revise different parts of this package. Bohmian mechanics adds an actual configuration and obtains effective subsystem collapse while the total wave evolves unitarily. The many-worlds interpretation retains the total superposition and treats each record as definite relative to its decohering branch. A stochastic collapse theory instead modifies the physical dynamics. Everett's original proposal explicitly separates his relative-state account from an additional collapse postulate.^[ref]

Thus collapse is not problematic merely because conditional probabilities change when a record is learned. The problem is the status of that record and of the dynamical rule used to describe it. A defensible interpretation must explain when its updating rule is applicable, what makes a record definite in its own ontology, and why the ordinary measurement statistics follow. No appeal to human consciousness is required by the displayed unitary measurement model.

Let $I=m{R}^2$, $T=t-t^{\prime }>0$, and $\Delta =\theta -{\theta }^{\prime }$. Periodic angular wave functions have modes $e^{il\theta }/\sqrt{2\pi }$ with integer $l$ and energies ${\hslash }^2{l}^2/(2I)$. Normalizing the kernel with respect to $d{\theta }^{\prime }$ givesIf the integration measure is arc length $Rd{\theta }^{\prime }$, the corresponding kernel is $K_{\theta }/R$. This measure choice accounts for the otherwise ambiguous radius prefactor.

In second quantization, expand the annihilating quantized field in a complete orthonormal basis of one-particle states, $\hat{\psi }(x)={\sum }_{\alpha }{\phi }_{\alpha }(x)a_{\alpha }$. At equal times, the canonical commutation relation for bosons and the canonical anticommutation relations for fermions are, respectively,andHere $\delta$ is a Dirac delta distribution. Multiplying by the basis wave functions and integrating gives $[a_{\alpha },a_{\beta }^{†}]={\delta }_{\alpha \beta }$ for bosons, or $\{a_{\alpha },a_{\beta }^{†}\}={\delta }_{\alpha \beta }$ for fermions; the two-annihilator and two-creator brackets vanish in the corresponding convention.

The mode number operator and total particle number operator areThe bosonic occupation numbers are $0,1,2,\dots$. For fermions, $a_{\alpha }^2=0$ and the canonical anticommutation relations imply $n_{\alpha }^2=n_{\alpha }$, so the fermionic occupation numbers are only $0$ and $1$, as required by the Pauli exclusion principle.

Keeping only two modes later is a projection: the projected field bracket is ${\sum }_{j=1}^2{\phi }_j(x){\phi }_j^{\ast }(x^{\prime })$, rather than the full Dirac delta distribution.

Away from the shell, the reduced radial equation is free. Regularity at the origin and the definition of the S-wave phase shift giveThe reduced radial wave function is continuous at $R$; otherwise its second derivative would contain a delta derivative not present in the equation. Integrating the equation across the shell gives the jumpDivide by the common shell value and substitute the two free solutions. This yieldsIf the shell value vanishes, use the undivided matching equations or a continuous limit rather than this logarithmic-derivative division. For an explicit phase shift of a delta-shell potential, write $x=kR$ and $g=2m{V}_{0}R/{\hslash }^2$. The cotangent addition formula givesThe phase is defined modulo $\pi$; choose its branch continuously as energy or coupling varies. At zero coupling the physical phase shift is zero.

A physicist who developed a relativistic theory of the electron. He shared the Nobel Prize in Physics in 1933 with Erwin Schrödinger.^[ref]

A two-component complex wave function describing the two spin states of a nonrelativistic spin-$1/2$ particle. Under a spatial rotation by angle $\theta$ about $\hat{\mathbf{n}}$ it transforms as $\psi \mapsto e^{-i\theta \hat{\mathbf{n}}\cdot \mathbf{\sigma }/2}\psi$, with Pauli matrices $\mathbf{\sigma }$. A Pauli spinor may describe a massive particle; two components alone do not imply the massless Weyl equation.

For $V(r)=V_0\delta (r-R)$, put $g=2m{V}_{0}R/{\hslash }^2$. The regular reduced radial wave function is proportional to $\sin kr$ inside the shell and to $\sin (kr+{\delta }_0)$ outside. It is continuous at $R$, while its radial derivative jumps by $(g/R)u(R)$. Dividing the jump condition by the common value gives $k\cot (kR+{\delta }_0)-k\cot (kR)=g/R$, which implies the displayed phase shift modulo $\pi$. At nodes, the undivided matching equations give the continuous limiting interpretation. For $g\ne 0$, an equivalent formula is $k\cot {\delta }_0=-k\cot (kR)-k(kR)/[g{\sin }^2(kR)]$. The exact S-wave scattering amplitude is $[k\cot {\delta }_0-ik{]}^{-1}$.

The material variables that a physical theory identifies with the distribution of stuff in ordinary space or spacetime, and through which it describes apparatus and recorded events. In particle Bohmian mechanics, actual particle configurations play this role; the wave function guides them. This is a way to specify an interpretation's account of matter, rather than a theorem that only particle ontologies are admissible.^[ref]

An ontological model of a quantum system is psi-epistemic if some pair of distinct pure states has preparation measures with nonzero overlap. The same physical state can then be compatible with either preparation. This precise measure-overlap definition is stronger than merely interpreting a wave function as information or a computational tool.

The ensemble density of actual configurations equals the squared norm of the guiding wave function. For spinless particles with $v_k=j_k/|\Psi {|}^2$, the Schrödinger continuity equation is ${\partial }_t|\Psi {|}^2+{\sum }_k{\nabla }_k\cdot (|\Psi {|}^2{v}_k)=0$. A density transported by the same guiding field satisfies the same equation, so initial equality is preserved. This equivariance is a consistency property, not a proof that every possible initial ensemble is in equilibrium. The physical account of the equilibrium assumption and subsystem statistics requires an additional argument.

The position-space kernel of the time-evolution operator propagates a wave function by $\psi (x,t)=\int K_U(x,t;x^{\prime },t^{\prime })\psi (x^{\prime },t^{\prime })d{x}^{\prime }$. It tends to $\delta (x-x^{\prime })$ as $t\to t^{\prime }$. A retarded convention multiplies it by the Heaviside step function: $K_R=\Theta (t-t^{\prime })K_U$. This convention obeys $(i\hslash {\partial }_t-H)K_R=i\hslash \delta (t-t^{\prime })\delta (x-x^{\prime })$; dividing by $i\hslash$ gives the Green function with a unit delta source. The time phase for an energy eigenstate is always $e^{-iE(t-t^{\prime })/\hslash }$.

A physicist.^[ref]

Coauthor of an ontological model of a quantum system classification distinguishing psi-ontic models and psi-epistemic models.

Coauthor of work on the GRW flash ontology and its relation to other quantum theories.

Uncertainty about which observer-centred situation is one's own, even conditional on a known global wave function. An example is a post-measurement observer who has not yet consulted a decohered record. This proposal distinguishes uncertainty about the experienced record from randomness in the universal dynamics. A further argument is needed to connect its credences to branch weights; not every decision-theoretic Everettian account must interpret its weights as ignorance of a uniquely selected future branch.

Spin-triplet p-wave pairing has pair spin $S=1$ and orbital angular momentum $L=1$, allowing multiple order parameters beyond conventional BCS theory with zero orbital angular momentum. ^[ref]

Specifying this symmetry does not determine its microscopic attraction mechanism. A short exchange-symmetry check explains why these quantum numbers fit fermions. The spin-one-half triplet states $|\uparrow \uparrow ⟩$, $(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)/\sqrt{2}$ and $|\downarrow \downarrow ⟩$ are symmetric under exchange. A stipulated orbital factor proportional to relative momentum $k_x$ changes sign under $\mathbf{k}\mapsto -\mathbf{k}$. Multiplying any symmetric spin function by that odd orbital factor gives an antisymmetric pair wave function, as required by Fermi-Dirac statistics. By comparison, an even s-wave orbital factor needs the antisymmetric spin singlet state. This checks compatibility with exchange statistics; it neither selects a stable phase nor derives its transition temperature.