FOUNDATIONS OF QUANTUM MECHANICS

128 CHAPTER VI. BOHMIAN MECHANICS
such a renunciation, but which instead leads us to regard a quantum - mechanical system
as a synthesis of a precisely definable particle and a precisely definable ψ - field which
exerts a force on this particle.
Bohm’s theory is strongly related to ideas which Louis de Broglie already put forward at the
Solvay Conference in 1927. However, criticism from the Copenhageners at the conference, especially
expressed by Pauli, made de Broglie abandon his theory, which was indeed not quite completely and
consistently developed. Bohm devised, independently of de Broglie, an entirely elaborated version,
which brought about a reconversion of de Broglie.
We will study Bohm’s theory because it is an example of a concrete HVT, in contrast to the abstract
characterization of such theories which we discussed in the previous chapter. We will see that Bohm’s
theory shows remarkable aspects which differ thoroughly from classical physics.
VI. 2
THE QUANTUM POTENTIAL
Bohm’s theory, which we will call Bohmian mechanics, starts from wave mechanics, i.e. quantum
mechanics with L 2 (R n ) as its Hilbert space, but without the projection postulate. 1 This means that
Bohm assumes that there is a wave function ψ(⃗q, t) which always satisfies the Schrödinger equation.
First we consider the 1 - particle case, if there are more particles, ψ has more arguments.
The idea is to interpret this wave function as a statistical description of a particle which always has
a certain position and momentum. We will see that this particle must then be subjected to dynamics
which differs from classical dynamics, by assuming that the forces acting on the particle are not
exclusively the forces known from classical physics.
The basic assumption is the Schrödinger equation for a particle with mass m in a time independent
potential V (⃗q),
i
∂ψ(⃗q, t)
∂t
= − 2
2 m ∇2 ψ(⃗q, t) + V (⃗q) ψ(⃗q, t), (VI. 1)
but we will interpret the wave function differently from its usual interpretation in quantum mechanics.
To this end, we rewrite ψ, with the help of two real functions R, S : R 4 → R, as
ψ(⃗q, t) = R(⃗q, t) e i S(⃗q, t) . (VI. 2)
It is always possible to find such functions R and S. Requiring R(⃗q, t) 0, R and S are, at given ψ,
uniquely defined, except where ψ = 0. Substitution of (VI. 2) in (VI. 1), and separating the real and
imaginary parts of the resulting equation, leads to two equations,
∂R(⃗q, t)
∂t
∂S(⃗q, t)
∂t
= − 1 (
R(⃗q, t) ∇ 2 S(⃗q, t) + 2 ∇ R(⃗q, t) · ∇ S(⃗q, t) ) ,
2 m
(VI. 3)
( ) 2 ∇ S(⃗q, t)
= −
− V (⃗q) +
2 ∇ 2 R(⃗q, t)
.
2 m
2 m R(⃗q, t)
(VI. 4)
1 In the literature, under Bohmian mechanics a ’streamlined’ version of Bohm’s original theory is understood, without a
quantum potential.