ABSTRACT
The solution is essentially a complex function which accounts for the matter wave spreading illustrated in Figure 28.3. As the wave function itself cannot be identified with a physical property of the free particle, an indirect interpretation was suggested by Born. He pointed out that its modulus
( tx,Ψ )
( ),x tΨ is large where the particle becomes localized and small elsewhere, and hence
( ) ( ) 2, ,P x t dx x t dx= Ψ (29.1)
is the probability of finding the particle described by ( )tx,Ψ at a point between x and at a time t. Therefore the matter wave spreading is associated with a decrease in
the probability density . As the total probability of finding the particle somewhere on the x-axis must be unity, the wave function must be normalized to unity, that is
dxx + ( txP , )
( ) ( ) ( ), , ,P x t dx x t x t dx∞ ∞ ∗ −∞ −∞
= Ψ Ψ∫ ∫ = (29.2)
The behaviour of the wave function ( )tx,Ψ is determined by Eq.(28.38), which may be rewritten as
( ) ( ) ( ) t x
tx m
itxttx ∆∂ Ψ∂+Ψ=∆+Ψ 2
2 , 2
,, = (29.3)
where the expression ( ) ( )/ , ,t x t t x t⎡∂Ψ ∂ ≅ Ψ + ∆ −Ψ ∆⎣ / t⎤⎦ )
, which holds for a small , has been used. In view of Eq.(29.3) the values of the wave function at any time can be derived from the initial value that describes a reference state of the particle.
