Computing the Fourier Transform
We find the expansion coefficients, A(q ) , by computing the Fourier transform of the
wave packet in x-space. The Fourier transform is defined in terms of continuous variables
x and q. To implement this in MATLAB® we need to approximate
+∞
A(q) = F (Ψ ( x)) = ∫ e − iqxΨ ( x)dx
−∞
by use of the discrete equation
F (Ψ ( x)) ≈ ∑ e−iqxΨ ( x)δ x
{ x}
To do this, we create a vector with all of our x values, i.e.:
⎛ x1 ⎞
⎜ ⎟
⎜ x2 ⎟
x = ⎜ x3 ⎟
⎜ ⎟
⎜ ⎟
⎜x ⎟
⎝ n⎠
Similarly, we define a vector with all the values of Ψ ( x ) :
⎛Ψ ( x1 ) ⎞
⎜
⎟
⎜Ψ ( x2 ) ⎟
Ψ ( x) = ⎜Ψ ( x3 ) ⎟
⎜
⎟
⎜
⎟
⎜Ψ ( x ) ⎟
n ⎠
⎝
We then define a vector for our reciprocal space:
⎛ q1 ⎞
⎜ ⎟
⎜ q2 ⎟
q = ⎜ q3 ⎟
⎜ ⎟
⎜ ⎟
⎜q ⎟
⎝ n⎠
The values for q are determined by the properties of the discrete Fourier transform. Note,
in general, x and q do not contain the same number of elements, i.e. m ≠ n .
Now, the outer product of x and q is
⎛ x1q1
⎜
xq
T
x⋅q = ⎜ 2 1
⎜
⎜
⎝ xn q1
x1q2 … x1qm ⎞
⎟
x2 q2 … x2 qm ⎟
⎟
⎟
xn q2
xn qm ⎠
Taking the exponential of i times each element of the above matrix yields:
φq ( x ) = e
ix ⋅qT
⎛ eix1q1
⎜ ix2 q1
e
=⎜
⎜
⎜⎜ ixn q1
⎝e
eix1q2
eix2 q2
eixn q2
eix1qm ⎞
⎟
eix2 qm ⎟
⎟
⎟
eixn qm ⎟⎠
Note, the MATLAB® command `` exp'' takes the exponent of each element in a matrix.
Then,
A(q ) = ∑ φq ( x)Ψ ( x)δ x
{ x}
= φ q† ( x) ⋅Ψ ( x)δ x
⎛ eix1q1
⎜ ix2 q1
e
=⎜
⎜
⎜⎜ ixn q1
⎝e
⎛ e −iq1x1
⎜ −iq2 x1
e
=⎜
⎜
⎜⎜ − iqm x1
⎝e
eix1q2
eix2 q2
eixn q2
e −iq1x2
e −iq2 x2
e − iqm x2
† ⎛Ψ ( x1 ) ⎞
eix1qm ⎞ ⎜
Ψ ( x2 ) ⎟⎟
ix2 qm ⎟ ⎜
e
⎟ ⋅ ⎜Ψ ( x ) ⎟ δ x
3
⎟ ⎜
⎟
⎟
⎜
⎟
eixn qm ⎟⎠ ⎜
⎟
⎝Ψ ( xn ) ⎠
⎛Ψ ( x1 ) ⎞
e − iq1xn ⎞ ⎜
Ψ ( x2 ) ⎟⎟
− iq2 xn ⎟ ⎜
e
⎟ ⋅ ⎜Ψ ( x ) ⎟ δ x
3
⎟ ⎜
⎟
⎟
⎜
⎟
e − iqm xn ⎟⎠ ⎜
⎟
⎝Ψ ( xn ) ⎠
⎛ Ψ ( x1 )e −iq1x1 +Ψ ( x2 )e− iq1x2 +
⎜
Ψ ( x1 )e− iq2 x1 +Ψ ( x2 )e− iq2 x2 +
⎜
=
⎜
⎜⎜
− iqm x1
+Ψ ( x2 )e − iqm x2 +
⎝Ψ ( x1 )e
+Ψ ( xn )e− iq1 xn ⎞
⎟
+Ψ ( xn )e− iq2 xn ⎟
δx
⎟
⎟
+Ψ ( xn )e− iqm xn ⎟⎠
i.e. a vector indexed by q which contains the sum we defined to be an approximation to
the Fourier transform, known as the Discrete Fourier Transform (DFT).