FOUNDATIONS OF QUANTUM MECHANICS

136 CHAPTER VI. BOHMIAN MECHANICS
Bohmian mechanics has not lead to new verifiable statements, although ‘tunneling times’ are debated,
about which quantum mechanics does not say anything, but Bohmian mechanics does. Furthermore,
by the fresh look supplied by Bohmian mechanics, new extensions of the theory are suggested,
such as the suggestion of an underlying sub - quantum aether, as a result of the unexpected double
role of the wave function.
In the nineteen nineties, a growing group of physicists considered Bohmian mechanics to be a
serious alternative for the Copenhagen interpretation, see for example Holland (1993) and Cushing
(1994), who suggests a sociological explanation for the fact that the physicists’ community did not
replace quantum mechanics by the, according to Cushing, superior Bohmian mechanics.
VI. 5
THE HAMILTON - JACOBI EQUATION
In classical mechanics we assume that for a system of n point particles, with canonical positions
⃗q = (q 1 , . . . , q n ) ∈ R 3n and speeds ˙⃗q = ( ˙q 1 , . . . , ˙q n ) ∈ R 3n , a Lagrangian L(⃗q, ˙⃗q, t) can be
found, the Lagrangian L = T − V being the difference between kinetic and potential energy. Define
the following functional, called the action
∫
S γ (⃗q, t; ⃗q 0 , t 0 ) := L(⃗q, ˙⃗q, t) dt, (VI. 29)
γ
where the integral, for n particles in 3 dimensions, is taken over a continuous path γ in configuration
space R 3n between an initial configuration ⃗q 0 at time t 0 and the configuration ⃗q at time t. In case the
Lagrangian does not explicitly depend on t, we can also write S γ (⃗q, ⃗q 0 , t − t 0 ).
The equations of motion are found by application of Hamilton’s principle of least action; for the
path γ 0 which is actually followed, the action reaches an extremum in comparison to all possible
continuous paths. This requirement,
δS γ = 0, (VI. 30)
provides n equations of motion of Euler and Lagrange,
d
dt
∂L
∂ ˙q j
− ∂L
∂q j
= 0. (VI. 31)
The Hamiltonian, H = T + V , is defined as the Legendre transform of the Lagrangian,
H (⃗q, ⃗p, t) :=
3n∑
j=1
p j ˙q j − L(⃗q, ˙⃗q, t) (VI. 32)
where
p j := ∂L
∂ ˙q j
(VI. 33)
is the canonical momentum.