18
2 The Discrete Spectrum and the Continuum
One demonstrates the existence and unicity of u(k, r) with the Picard method by
rewriting Eqs. (2.2) and (2.6) as an integral equation:
u(k, r) = C 0 F 0 ,η 0 (k 0 r 0 ) + C 0 k 0 F
0 ,η 0
(k 0 r 0 ) (r − r 0 )
(2.8)
+
r
r 0
(r − r
)L (k, r
)u(k, r
) dr
,
with r ≥ r 0 . In this equation, L (k, r) is an operator representing the action of
potentials and linear momentum in Eq. (2.2):
L (k, r) =
+ 1)
r 2
+ v l (r) − k
2 .
(2.9)
The demonstration is effected in Exercise I. It is, however, rather technical, so that
readers can omit it if they find it too difficult and directly assume that u(k, r) exists
and is unique.
Exercise I
In this exercise, one will show that Eq. (2.2) has a unique solution for every
linear momentum k.
A. Define u (n+1) (k, r) as a function of u (n) (k, r) from Eq. (2.9), so as to obtain
a recurrence relation between the two functions. In order to fix normalization
constants, one will pose C 0 = k
− 0 −1
0
C 0 (η 0 ) −1 in Eqs. (2.6) and (2.9).
Deduce from the integral equation verified by u (n) (k, r) that u(k, r) exists and
that u (n) (k, r) → u(k, r) uniformly in every finite domain of the complex
k-plane.
B. Show that the solution of Eq. (2.2) built from Eq. (2.9) is unique.
For this, let us consider the Δu(k, r) function, difference of two solutions of
Eq. (2.9) whose functions and derivatives are equal in one point.
Using the same recurrence relation scheme as in A, show that Δu(k, r) = 0
and conclude that u(k, r) is unique.
2.2
The Harmonic Oscillator Potential
The harmonic oscillator Hamiltonian and wave functions pervade all quantum
mechanics. It is the simplest nontrivial Hamiltonian to bear analytical solutions
and related to many physical processes. Similarly to its classical analog, harmonic
vibration around equilibrium is the most basic motion, and can describe a large
variety of physical systems at low energy. Moreover, the harmonic oscillator
Hamiltonian is analytical for all its eigenstates, which is very rare in quantum
mechanics.
2 The Discrete Spectrum and the Continuum
One demonstrates the existence and unicity of u(k, r) with the Picard method by
rewriting Eqs. (2.2) and (2.6) as an integral equation:
u(k, r) = C 0 F 0 ,η 0 (k 0 r 0 ) + C 0 k 0 F
0 ,η 0
(k 0 r 0 ) (r − r 0 )
(2.8)
+
r
r 0
(r − r
)L (k, r
)u(k, r
) dr
,
with r ≥ r 0 . In this equation, L (k, r) is an operator representing the action of
potentials and linear momentum in Eq. (2.2):
L (k, r) =
+ 1)
r 2
+ v l (r) − k
2 .
(2.9)
The demonstration is effected in Exercise I. It is, however, rather technical, so that
readers can omit it if they find it too difficult and directly assume that u(k, r) exists
and is unique.
Exercise I
In this exercise, one will show that Eq. (2.2) has a unique solution for every
linear momentum k.
A. Define u (n+1) (k, r) as a function of u (n) (k, r) from Eq. (2.9), so as to obtain
a recurrence relation between the two functions. In order to fix normalization
constants, one will pose C 0 = k
− 0 −1
0
C 0 (η 0 ) −1 in Eqs. (2.6) and (2.9).
Deduce from the integral equation verified by u (n) (k, r) that u(k, r) exists and
that u (n) (k, r) → u(k, r) uniformly in every finite domain of the complex
k-plane.
B. Show that the solution of Eq. (2.2) built from Eq. (2.9) is unique.
For this, let us consider the Δu(k, r) function, difference of two solutions of
Eq. (2.9) whose functions and derivatives are equal in one point.
Using the same recurrence relation scheme as in A, show that Δu(k, r) = 0
and conclude that u(k, r) is unique.
2.2
The Harmonic Oscillator Potential
The harmonic oscillator Hamiltonian and wave functions pervade all quantum
mechanics. It is the simplest nontrivial Hamiltonian to bear analytical solutions
and related to many physical processes. Similarly to its classical analog, harmonic
vibration around equilibrium is the most basic motion, and can describe a large
variety of physical systems at low energy. Moreover, the harmonic oscillator
Hamiltonian is analytical for all its eigenstates, which is very rare in quantum
mechanics.