Momentum Space Wave functions:
Let ψ(x) be the wavefunction of a one-dimensional system and let φ(kx ) ≡ φ(px ) be its
Fourier transform in the k space. We know that
φ(px ) =
1 Z +∞
ψ( x)eikx x dkx
(2π)1/2 −∞
Z +∞
px x
1
px
ψ( x)ei h̄ dpx as kx =
1/2
(2πh̄)
h̄
−∞
(1)
The three dimensional analogue of the above equation can be written as
φ(p) =
1 Z +∞ Z +∞ Z +∞
ψ(r)eik·r dkx dky dkz
(2π)3/2 −∞ −∞ −∞
Z +∞ Z +∞ Z +∞
p·r
1
ψ( r)ei h̄ dpx dpy dpz .
3/2
(2πh̄)
−∞
−∞
−∞
(2)
It is not difficult to notice that the function
ψp (r) ≡
1
i p·r
h̄
e
(2πh̄)3/2
(3)
~ In fact, as
is the normalized eigen function of the momentum operator p̂ = −ih̄∇.
p̂x ψpx (x) = −ih̄
ipx x
∂ ψpx (x)
1
= px
e h̄ = px ψpx (x),
1/2
∂x
(2πh̄)
the three dimensional analogue of the eigenvalue equation follows immediately. That is,
~ p (r) = pψp (r)
p̂ ψp (r) = −ih̄∇ψ
(4)
A convenient notation for the momentum space wave functions is given by ψp (r).
Also, as the function φ(p) is the coefficient of expansion of the co-ordinate space wavefunction in terms of the momentum eigenfunctions. we can say that φ(p) is the wavefunction
(of the same physical system represented by the coordinate space wave function ψ(r)) in
momentum space. In short, we say that φ(p) is the momentum-space wavefunction corresponding to ψ(r).
1