ABSTRACT
According to non-relativistic quantummechanics, the evolution of the wave function of a free particle of mass m and momentum p is described by the Schro¨dinger equation:
i ∂ψ
∂t = Hψ , (6.1)
with the Hamiltonian operator H = −∇2/2m obtained from the expression for energy
E = p2
2m , (6.2)
using the substitution
E → i ∂ ∂t
, p → −i∇ . (6.3) Schro¨dinger himself [6] first suggested a generalisation of equation (6.1) based on the use of the relativistic energy equation:
E2 = p2 +m2 . (6.4)
(+m2)ψ = 0 , (6.5)
Multiplying by ψ∗ and subtracting the product of ψ and the complex conjugate of (6.5) from the result, the continuity equation is obtained
∂ρ
∂t = ∇j , (6.6)
with
ρ = ψ∗ ∂ψ
∂t − ψ∂ψ
∂t (6.7)
and j = −∇ (ψ∗∇ψ − ψ∇ψ∗) . (6.8)
However, the ρ which appears in equation (6.6) cannot be identified with the probability density, in analogy with the similar quantity obtained from the Schro¨dinger equation, because it does not have the required property of being always positive-definite. To be convinced of this it is sufficient to substitute i∂/∂t→ E in (6.7). The result,
ρ = E|ψ|2 , shows that ρ can be either positive or negative, following from the fact that the Klein-Gordon equation has solutions for the energy with both signs
E = ± √ p2 +m2 ,
Note that in the non-relativistic limit E ≈ m > 0, and the familiar result ρ ∝ |ψ|2 is recovered.
