Quantum potential

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. Q = − ℏ 2 2 m ∇ 2 R R . {displaystyle Q=-{frac {hbar ^{2}}{2m}}{frac { abla ^{2}R}{R}}.} 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. 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 formally more complete exposition of the idea presented by Louis de Broglie: de Broglie had postulated in 1926 that the wave function represents a pilot wave which guides a quantum particle, but had subsequently abandoned his approach due to objections raised by Wolfgang Pauli. 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 work by Erwin Madelung of 1927 and to work by Carl Friedrich von Weizsäcker of 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. The Schrödinger equation is re-written using the polar form for the wave function ψ = R exp ⁡ ( i S / ℏ ) {displaystyle psi =Rexp(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.