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Equation 3.6, also called the “time independent Schrödinger equation”, frequently
has many possible solutions for different values of E. These solutions, the
eigenvectors of Ĥ, are specific to each system and have physical meaning in that
they describe the possible pure quantum states that the system can exist in. They
are therefore called the system’s “eigenstates”. The eigenvalue E corresponding
to an eigenstate gives its energy and the eigenstate with the lowest eigenvalue (i.e.
lowest energy) describes the ground-state into which the system will relax if it is
cooled sufficiently (provided it does not get trapped in a local energy minimum).
Commonly, the states are sorted by ascending energy and labeled with an index
starting at 0. The n th eigenstate ψn r (r) has energy E n and is written as | n 〉. The
full solution to the Schrödinger equation for the n th eigenstate is:
ψ (r, t) = ψ r (r) ψ t (t) = e −iEnt/ ψ r (r) = e −iEnt/ | n 〉 (3.8)
Since the eigenstates form a complete basis, any possible real state ψ (r) (or | ψ 〉
for short) that the system might exist in can be written as a linear superposition
of eigenstates, i.e.:
| ψ 〉 = ∑ n
a n | n 〉 (3.9)
The coefficients a n are calculated by projection:
∫
a n =
ψ (r) ∗ ψ r n (r) dr = 〈 ψ | n 〉 (3.10)
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