The quantum potential or quantum potentiality is a central concept of the de Broglie–Bohm formulation of quantum mechanics, introduced by David Bohm in 1952. Initially presented under the name quantum-mechanical potential, subsequently quantum potential, it was later elaborated upon by Bohm and Basil Hiley in its interpretation as an information potential which acts on a quantum particle. It is also referred to as quantum potential energy, Bohm potential, quantum Bohm potential or Bohm quantum potential.
In the framework of the de Broglie–Bohm theory, the quantum potential is a term within the Schrödinger equation which acts to guide the movement of quantum particles. The quantum potential approach introduced by Bohm provides a physically less fundamental exposition of the idea presented by Louis de Broglie: de Broglie had postulated in 1925 that the relativistic wave function defined on spacetime represents a pilot wave which guides a quantum particle, represented as an oscillating peak in the wave field, but he had subsequently abandoned his approach because he was unable to derive the guidance equation for the particle from a non-linear wave equation. The seminal articles of Bohm in 1952 introduced the quantum potential and included answers to the objections which had been raised against the pilot wave theory. The Bohm quantum potential is closely linked with the results of other approaches, in particular relating to works of Erwin Madelung in 1927 and Carl Friedrich von Weizsäcker in 1935. Building on the interpretation of the quantum theory introduced by Bohm in 1952, David Bohm and Basil Hiley in 1975 presented how the concept of a quantum potential leads to the notion of an "unbroken wholeness of the entire universe", proposing that the fundamental new quality introduced by quantum physics is nonlocality.
Relation to the Schrödinger equation
The Schrödinger equation
i ℏ ∂ ψ ∂ t = ( − ℏ 2 2 m ∇ 2 + V ) ψ {\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\right)\psi \quad }
is re-written using the polar form for the wave function ψ = R exp ( i S / ℏ ) {\displaystyle \psi =R\exp(iS/\hbar )} with real-valued functions R {\displaystyle R} and S {\displaystyle S} , where R {\displaystyle R} is the amplitude (absolute value) of the wave function ψ {\displaystyle \psi } , and S / ℏ {\displaystyle S/\hbar } its phase. This yields two equations: from the imaginary and real part of the Schrödinger equation follow the continuity equation and the quantum Hamilton–Jacobi equation respectively.
Continuity equation The imaginary part of the Schrödinger equation in polar form yields
∂ R ∂ t = − 1 2 m [ R ∇ 2 S + 2 ∇ R ⋅ ∇ S ] , {\displaystyle {\frac {\partial R}{\partial t}}=-{\frac {1}{2m}}\left[R\nabla ^{2}S+2\nabla R\cdot \nabla S\right],}
which, provided ρ = R 2 {\displaystyle \rho =R^{2}} , can be interpreted as the continuity equation ∂ ρ / ∂ t + ∇ ⋅ ( ρ v ) = 0 {\displaystyle \partial \rho /\partial t+\nabla \cdot (\rho v)=0} for the probability density ρ {\displaystyle \rho } and the velocity field v = 1 m ∇ S {\displaystyle v={\frac {1}{m}}\nabla S}
Quantum Hamilton–Jacobi equation The real part of the Schrödinger equation in polar form yields a modified Hamilton–Jacobi equation also referred to as quantum Hamilton–Jacobi equation. It differs from the classical Hamilton–Jacobi equation only by the term This term Q {\displaystyle Q} , called quantum potential, thus depends on the curvature of the amplitude of the wave function. In the limit ℏ → 0 {\displaystyle \hbar \to 0} , the function S {\displaystyle S} is a solution of the (classical) Hamilton–Jacobi equation; therefore, the function S {\displaystyle S} is also called the Hamilton–Jacobi function, or action, extended to quantum physics.
Properties
Hiley emphasised several aspects that regard the quantum potential of a quantum particle:
it is derived mathematically from the real part of the Schrödinger equation under polar decomposition of the wave function, is not derived from a Hamiltonian or other external source, and could be said to be involved in a self-organising process involving a basic underlying field; it does not change if R {\displaystyle R} is multiplied by a constant, as this term is also present in the denominator, so that Q {\displaystyle Q} is independent of the magnitude of ψ {\displaystyle \psi } and thus of field intensity; therefore, the quantum potential fulfils a precondition for nonlocality: it need not fall off as distance increases; it carries information about the whole experimental arrangement in which the particle finds itself. In 1979, Hiley and his co-workers Philippidis and Dewdney presented a full calculation on the explanation of the two-slit experiment in terms of Bohmian trajectories that arise for each particle moving under the influence of the quantum potential, resulting in the well-known interference patterns.
Also the shift of the interference pattern which occurs in presence of a magnetic field in the Aharonov–Bohm effect could be explained as arising from the quantum potential.
Relation to the measurement process The collapse of the wave function of the Copenhagen interpretation of quantum theory is explained in the quantum potential approach by the demonstration that, after a measurement, "all the packets of the multi-dimensional wave function that do not correspond to the actual result of measurement have no effect on the particle" from then on. Bohm and Hiley pointed out that
...the quantum potential can develop unstable bifurcation points, which separate classes of particle trajectories according to the "channels" into which they eventually enter and within which they stay. This explains how measurement is possible without "collapse" of the wave function, and how all sorts of quantum processes, such as transitions between states, fusion of two states into one and fission of one system into two, are able to take place without the need for a human observer. Measurement then "involves a participatory transformation in which both the system under observation and the observing apparatus undergo a mutual participation so that the trajectories behave in a correlated manner, becoming correlated and separated into different, non-overlapping sets (which we call 'channels')".
Quantum potential of an n-particle system The Schrödinger wave function of a many-particle quantum system cannot be represented in ordinary three-dimensional space. Rather, it is represented in configuration space, with three dimensions per particle. A single point in configuration space thus represents the configuration of the entire n-particle system as a whole. A two-particle wave function ψ ( r 1 , r 2 , t ) {\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)} of identical particles of mass m {\displaystyle m} has the quantum potential
Q ( r 1 , r 2 , t ) = − ℏ 2 2 m ( ∇ 1 2 + ∇ 2 2 ) R ( r 1 , r 2 , t ) R ( r 1 , r 2 , t ) {\displaystyle Q(\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {(\nabla _{1}^{2}+\nabla _{2}^{2})R(\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)}{R(\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)}}}
where ∇ 1 2 {\displaystyle \nabla _{1}^{2}} and ∇ 2 2 {\displaystyle \nabla _{2}^{2}} refer to particle 1 and particle 2 respectively. This expression generalizes in straightforward manner to n {\displaystyle n} particles:
Q ( r 1 , . . . , r n , t ) = − ℏ 2 2 R ( r 1 , . . . , r n , t ) ∑ i = 1 n ∇ i 2 m i R ( r 1 , . . . , r n , t ) {\displaystyle Q(\mathbf {r_{1}} ,...,\mathbf {r_{n}} ,\,t)=-{\frac {\hbar ^{2}}{2R(\mathbf {r_{1}} ,...,\mathbf {r_{n}} ,\,t)}}\sum _{i=1}^{n}{\frac {\nabla _{i}^{2}}{m_{i}}}R(\mathbf {r_{1}} ,...,\mathbf {r_{n}} ,\,t)}
In case the wave function of two or more particles is separable, then the system's total quantum potential becomes the sum of the quantum potentials of the two particles. Exact separability is extremely unphysical given that interactions between the system and its environment destroy the factorization; however, a wave function that is a superposition of several wave functions of approximately disjoint support will factorize approximately.
Derivation for a separable quantum system That the wave function is separable means that ψ {\displaystyle \psi } factorizes in the form ψ ( r 1 , r 2 , t ) = ψ A ( r 1 , t ) ψ B ( r 2 , t ) {\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=\psi _{A}(\mathbf {r_{1}} ,\,t)\psi _{B}(\mathbf {r_{2}} ,\,t)} . Then it follows that also R {\displaystyle R} factorizes, and the system's total quantum potential becomes the sum of the quantum potentials of the two particles.
Q ( r 1 , r 2 , t ) = − ℏ 2 2 m ( ∇ 1 2 R A ( r 1 , t ) R A ( r 1 , t ) + ∇ 2 2 R B ( r 2 , t ) R B ( r 2 , t ) ) = Q A ( r 1 , t ) + Q B ( r 2 , t ) {\displaystyle Q(\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=-{\frac {\hbar ^{2}}{2m}}({\frac {\nabla _{1}^{2}R_{A}(\mathbf {r_{1}} ,\,t)}{R_{A}(\mathbf {r_{1}} ,\,t)}}+{\frac {\nabla _{2}^{2}R_{B}(\mathbf {r_{2}} ,\,t)}{R_{B}(\mathbf {r_{2}} ,\,t)}})=Q_{A}(\mathbf {r_{1}} ,\,t)+Q_{B}(\mathbf {r_{2}} ,\,t)}
In case the wave function is separable, that is, if ψ {\displaystyle \psi } factorizes in the form ψ ( r 1 , r 2 , t ) = ψ A ( r 1 , t ) ψ B ( r 2 , t ) {\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=\psi _{A}(\mathbf {r_{1}} ,\,t)\psi _{B}(\mathbf {r_{2}} ,\,t)} , the two one-particle systems behave independently. More generally, the quantum potential of an n {\displaystyle n} -particle system with separable wave function is the sum of n {\displaystyle n} quantum potentials, separating the system into n {\displaystyle n} independent one-particle systems.
Formulation in terms of probability density
Quantum potential in terms of the probability density function Bohm, as well as other physicists after him, have sought to provide evidence that the Born rule linking R {\displaystyle R} to the probability density function
ρ = R 2 {\displaystyle \rho =R^{2}\quad }
can be understood, in a pilot wave formulation, as not representing a basic law, but rather a theorem (called quantum equilibrium hypothesis) which applies when a quantum equilibrium is reached during the course of the time development under the Schrödinger equation. With Born's rule, and straightforward application of the chain and product rules
∇ 2 ρ = ∇ ∇ ρ 1 / 2 = ∇ ( 1 2 ρ − 1 / 2 ∇ ρ ) = 1 2 [ ( ∇ ρ − 1 / 2 ) ∇ ρ + ρ − 1 / 2 ∇ 2 ρ ] {\displaystyle \nabla ^{2}{\sqrt {\rho }}=\nabla \nabla \rho ^{1/2}=\nabla \left({\frac {1}{2}}\rho ^{-1/2}\nabla \rho \right)={\frac {1}{2}}\left[\left(\nabla \rho ^{-1/2}\right)\nabla \rho +\rho ^{-1/2}\nabla ^{2}\rho \right]}
the quantum potential, expressed in terms of the probability density function, becomes:
Q = − ℏ 2 2 m ∇ 2 ρ ρ = − ℏ 2 4 m [ ∇ 2 ρ ρ − 1 2 ( ∇ ρ ) 2 ρ 2 ] {\displaystyle Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}{\sqrt {\rho }}}{\sqrt {\rho }}}=-{\frac {\hbar ^{2}}{4m}}\left[{\frac {\nabla ^{2}\rho }{\rho }}-{\frac {1}{2}}{\frac {(\nabla \rho )^{2}}{\rho ^{2}}}\right]}
Quantum force The quantum force F Q = − ∇ Q {\displaystyle F_{Q}=-\nabla Q} , expressed in terms of the probability distribution, amounts to:
F Q = ℏ 2 4 m [ ∇ ( ∇ 2 ρ ) ρ − ∇ ( ∇ ρ ⋅ ∇ ρ ) 2 ρ 2 − ( ∇ 2 ρ ρ − ∇ ρ ⋅ ∇ ρ ρ 2 ) ∇ ρ ρ ] {\displaystyle F_{Q}={\frac {\hbar ^{2}}{4m}}\left[{\frac {\nabla (\nabla ^{2}\rho )}{\rho }}-{\frac {\nabla (\nabla \rho \cdot \nabla \rho )}{2\rho ^{2}}}-\left({\frac {\nabla ^{2}\rho }{\rho }}-{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}\right){\frac {\nabla \rho }{\rho }}\right]}
Formulation in configuration space and in momentum space, as the result of projections M. R. Brown and B. Hiley showed that, as alternative to its formulation terms of configuration space ( x {\displaystyle x} -space), the quantum potential can also be formulated in terms of momentum space ( p {\displaystyle p} -space). In line with David Bohm's approach, Basil Hiley and mathematician Maurice de Gosson showed that the quantum potential can be seen as a consequence of a projection of an underlying structure, more specifically of a non-commutative algebraic structure, onto a subspace such as ordinary space ( x {\displaystyle x} -space). In algebraic terms, the quantum potential can be seen as arising from the relation between implicate and explicate orders: if a non-commutative algebra is employed to describe the non-commutative structure of the quantum formalism, it turns out that it is impossible to define an underlying space, but that rather "shadow spaces" (homomorphic spaces) can be constructed and that in so doing the quantum potential appears. The quantum potential approach can be seen as a way to construct the shadow spaces. The quantum potential thus results as a distortion due to the projection of the underlying space into x {\displaystyle x} -space, in similar manner as a Mercator projection inevitably results in a distortion in a geographical map. There exists complete symmetry between the x {\displaystyle x} -representation, and the quantum potential as it appears in configuration space can be seen as arising from the dispersion of the momentum p {\displaystyle p} -representation. The approach has been applied to extended phase space, also in terms of a Duffin–Kemmer–Petiau algebra approach.
Relation to other quantities and theories
Relation to the Fisher information It can be shown that the mean value of the quantum potential Q = − ℏ 2 ∇ 2 ρ / ( 2 m ρ ) {\displaystyle Q=-\hbar ^{2}\nabla ^{2}{\sqrt {\rho }}/(2m{\sqrt {\rho }})} is proportional to the probability density's Fisher information about the observable x ^ {\displaystyle {\hat {x}}}
I = ∫ ρ ⋅ ( ∇ ln ρ ) 2 d 3 x = − ∫ ρ ∇ 2 ( ln ρ ) d 3 x . {\displaystyle {\mathcal {I}}=\int \rho \cdot (\nabla \ln \rho )^{2}\,d^{3}x=-\int \rho \nabla ^{2}(\ln \rho )\,d^{3}x.}
Using this definition for the Fisher information, we can write:
⟨ Q ⟩ = ∫ ψ ∗ Q ψ d 3 x = ∫ ρ Q d 3 x = ℏ 2 8 m I . {\displaystyle \langle Q\rangle =\int \psi ^{*}Q\psi \,d^{3}x=\int \rho Q\,d^{3}x={\frac {\hbar ^{2}}{8m}}{\mathcal {I}}.}
Quantum potential as energy of internal motion associated with spin Giovanni Salesi, Erasmo Recami and co-workers showed in 1998 that, in agreement with the König's theorem, the quantum potential can be identified with the kinetic energy of the internal motion ("zitterbewegung") associated with the spin of a spin-1/2 particle observed in a center-of-mass frame. More specifically, they showed that the internal zitterbewegung velocity for a spinning, non-relativistic particle of constant spin with no precession, and in absence of an external field, has the squared value:
V 2 = ( ∇ ρ ∧ s ) 2 ( m ρ ) 2 = ( ∇ ρ ) 2 s 2 − ( ∇ ρ ⋅ s ) 2 ( m ρ ) 2 {\displaystyle \mathbf {V} ^{2}={\frac {(\nabla \rho \land \mathbf {s} )^{2}}{(m\rho )^{2}}}={\frac {(\nabla \rho )^{2}\mathbf {s} ^{2}-(\nabla \rho \cdot \mathbf {s} )^{2}}{(m\rho )^{2}}}}
from which the second term is shown to be of negligible size; then with | s | = ℏ / 2 {\displaystyle |\mathbf {s} |=\hbar /2} it follows that
| V | = ℏ 2 ∇ ρ m ρ {\displaystyle |\mathbf {V} |={\frac {\hbar }{2}}{\frac {\nabla \rho }{m\rho }}}
Salesi gave further details on this work in 2009. In 1999, Salvatore Esposito generalized their result from spin-1/2 particles to particles of arbitrary spin, confirming the interpretation of the quantum potential as a kinetic energy for an internal motion. Esposito showed that (using the notation ℏ {\displaystyle \hbar } =1) the quantum potential can be written as:
Q = − 1 2 m v S 2 − 1 2 ∇ ⋅ v S {\displaystyle Q=-{\frac {1}{2}}m\mathbf {v} _{S}^{2}-{\frac {1}{2}}\nabla \cdot \mathbf {v} _{S}}
and that the causal interpretation of quantum mechanics can be reformulated in terms of a particle velocity
v = v B + v S × s {\displaystyle \mathbf {v} =\mathbf {v} _{B}+\mathbf {v} _{S}\times \mathbf {s} }
where the "drift velocity" is
v B = ∇ S m {\displaystyle \mathbf {v} _{B}={\frac {\nabla S}{m}}}
and the "relative velocity" is v S × s {\displaystyle \mathbf {v} _{S}\times \mathbf {s} } , with
v S = ∇ R 2 2 m R 2 {\displaystyle \mathbf {v} _{S}={\frac {\nabla R^{2}}{2mR^{2}}}}
and s {\displaystyle \mathbf {s} } representing the spin direction of the particle. In this formulation, according to Esposito, quantum mechanics must necessarily be interpreted in probabilistic terms, for the reason that a system's initial motion condition cannot be exactly determined. Esposito explained that "the quantum effects present in the Schrödinger equation are due to the presence of a peculiar spatial direction associated with the particle that, assuming the isotropy of space, can be identified with the spin of the particle itself". Esposito generalized it from matter particles to gauge particles, in particular photons, for which he showed that, if modelled as ψ = ( E − i B ) / 2 {\displaystyle \psi =(\mathbf {E} -i\mathbf {B} )/{\sqrt {2}}} , with probability function ψ ∗ ⋅ ψ = ( E 2 + B 2 ) / 2 {\displaystyle \psi ^{*}\cdot \psi =(\mathbf {E} ^{2}+\mathbf {B} ^{2})/2} , they can be understood in a quantum potential approach. James R. Bogan, in 2002, published the derivation of a reciprocal transformation from the Hamilton-Jacobi equation of classical mechanics to the time-dependent Schrödinger equation of quantum mechanics which arises from a gauge transformation representing spin, under the simple requirement of conservation of probability. This spin-dependent transformation is a function of the quantum potential.
Re-interpretation in terms of Clifford algebras B. Hiley and R. E. Callaghan re-interpret the role of the Bohm model and its notion of quantum potential in the framework of Clifford algebra, taking account of recent advances that include the work of David Hestenes on spacetime algebra. They show how, within a nested hierarchy of Clifford algebras C ℓ i , j {\displaystyle C\ell _{i,j}} , for each Clifford algebra an element of a minimal left ideal Φ L ( r , t ) {\displaystyle \Phi _{L}(\mathbf {r} ,t)} and an element of a right ideal representing its Clifford conjugation