FOUNDATIONS OF QUANTUM MECHANICS

130 CHAPTER VI. BOHMIAN MECHANICS
EXAMPLE
A particle sits in a 1 - dimensional ‘box’ of length L, having walls which are formed by infinitely
high potential barriers. Quantum mechanics gives as stationary solutions
ψ n (q, t) = ψ n (q) e − i En t , (VI. 9)
with
ψ n (q) =
√
2
( nπq
)
L sin , q ∈ [0, L], (VI. 10)
L
and energy values
E n =
2
2 m
( n π
) 2
. (VI. 11)
L
Therefore, in Bohmian mechanics for a stationary state we have
R n (q, t) = ψ n (q) and S n (q, t) = − E n t. (VI. 12)
Now it is surprising that in this example it holds that
p = ∂S n
∂q
= ∂(− E n t)
∂q
= 0, (VI. 13)
i.e., according to Bohmian mechanics the particle is motionless. This also applies to other cases of
stationary states, for example to the ground state of the hydrogen atom. It is in straight contradiction
to the statements of quantum mechanics. After all, in the case of the box quantum mechanics
assigns, if the particle is in the state ψ n , a large probability to finding the momentum p having values
around ±nπ
L
, in which case the particle moves with p m
> 0, although the quantum mechanical
expectation value of p is zero for the particle in the box.
This example shows that the statements of quantum mechanics and Bohmian mechanics do not
coincide for all quantities. They only correspond concerning probability distributions for position
measurements. Bohmian mechanics is, therefore, not a HVT in the sense of chapter V, where it was
assumed that the statements of such a theory are similar to the statements of quantum mechanics for
all quantities. Von Neumann’s impossibility proof is therefore not applicable to Bohmian mechanics.
The explanation of the discrepancy between Bohmian mechanics and quantum mechanics lies, of
course, in the use of the quantum potential. According to Bohm, the energy of the particle in the box
has been entirely stored in the form of potential energy as a result of the quantum potential, hence,
the particle has no kinetic energy.
This changes however as soon as we open the box by removing one or both barriers. The quantum
potential energy is again released, and the particle will start to move. The wave packet ψ(⃗q, t) then
spreads out in space, in exactly the same way as prescribed by the Schrödinger equation, and there
is no difference anymore between the statements of both theories concerning the movement of the
particle.