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As the eigenstates are ortho-normal, 〈 n | m 〉 = 1 if n = m, otherwise 〈 n | m 〉 = 0:
(
i ∂ t an (t) e ) −iE nt/
=
∑
a m (t) e −iEmt/ E m 〈 n | m 〉 + ∑
m
m
a m (t) e −iE mt/ 〈 n | V (t) | m 〉
i e −iE nt/ ∂ t a n (t) + E n a n (t) e −iE nt/ =
E n a n (t) e −iEnt/ + ∑ m
a m (t) e −iEmt/ 〈 n | V (t) | m 〉
i ∂ t a n (t) = ∑ m
a m (t) e i(En−Em)t/ 〈 n | V (t) | m 〉 (3.14)
V (t) is often a function of r, in which case:
∫
〈 n | V (r, t) | m 〉 =
ψ r n (r) ∗ V (r, t) ψ r m (r) dr (3.15)
3.2 Finding Eigenstates Numerically
The first step in understanding the qubit is to find its eigenstates. The qubit
behaves like a particle moving along a single axis x in a potential V (x). Its total
energy is given by the particle’s kinetic energy T and its potential energy V , i.e.:
Ĥ = T + V = − 2
2m
d 2
+ V (x) (3.16)
dx2 With this, the time independent Schrödinger equation becomes:
E n ψ r n(x) = − 2
2m
d 2
dx 2 ψr n(x) + V (x) ψ r n(x) (3.17)
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