PDF (double-sided) - Physics Department, UCSB - University of ...

This can be expressed as a tri-diagonal matrix operating on the vector ψ r n, here:
⎡
d 2
dx = D = 2 ⎢
⎢
⎣
−2 1
1 −2 1
1 −2 1
1 −2 1
1 −2 1
1 −2 1
1 −2
⎤
⎥
⎦
(3.21)
Note that if x is discretized in steps of dx ≠ 1, D will need to be divided by
dx 2 . Now we can rewrite the time independent Schrödinger equation as a matrix
equation:
E n ψ r n =
(− 2
2m D + I V )
ψ r n (3.22)
where I in this case is the 7 × 7 identity. To solve this equation, one needs to
find the eigenvectors of the matrix M = (I V − 2 D). This can be done using
2m
the “eig” function of the “LAPACK” software routines, e.g. via Matlab (eig) or
Python (numpy.linalg.eig). For m = 2 , M in our example becomes:
4
⎡
M = (I V − 2D) =
⎢
⎣
13 −2
−2 8 −2
−2 5 −2
−2 4 −2
−2 5 −2
−2 8 −2
−2 13
⎤
⎥
⎦
(3.23)
The eigenvalues of this matrix are:
1.35, 3.91, 6.04, 8.34, 8.85, 13.75, 13.76
43