The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 52

Since the set{|ψ r }represents a basis, it can be employed to expand the

complete Schrodinger state vector|Υ k of H from (2.45). Furthermore, the

total state vector can be written at any energy E, and not only E k , through
the following development

∞

B n (E)|ψ (s)

|Υ(E)= . (2.108)

n

n=0

, which is present in|ψ (s) This is valid for any vector|Φ s through the state

n

= Q n ( H)|Φ s |ψ (s) at any energy

E. Moreover, the following expression for the expansion coe cients B n (E)
can be obtained

. Therefore,|Υ(E)is independent of|Φ s

n

B n (E) = W n (E)Q n (E). (2.109)

As such, the exact Schrodinger eigenvector|Υ(E)of H can be generated by
propagation of the state|Φ s

|Υ(E)=P(E, H)|Φ s

(2.110)

∞

P(E, H) = W n (E)Q n (E)Q n ( H). (2.111)

n=0

For example, the vector|Φ s can be viewed as the initial state of the studied
system. More generally, an arbitrary state can be used for|Φ s including a
random state. Comparison of the completeness relation (2.37) with (2.111)
gives the relation

|Υ(E)= δ(E1−H)|Φ s . (2.112)

The eigenstates|Υ k ≡|Υ(E k ), associated with the eigenvalue E k , can be

extracted from the expansion (2.108) taken at E = E k

∞ ∞

W n (E k )Q n (E k )Q n ( H )|Φ s B n (E k )|ψ (s)

|Υ k = = . (2.113)

n

n=0 n=0

By way of a summary, let us enumerate the key properties of the expansion
method devised using nonclassical polynomials that are intertwined with ra
tional polynomials as analyzed in this chapter (see also [127, 128]):

•Diagonal and offdiagonal elements can be computed, not only for the
Green function itself, but also for the product of any number of Green
functions with other quantummechanical operators,

•The energy resolution in the sought spectrum is controlled by the ex
pansion order, which is equal to the number of retained polynomials,