tom.stoer said:
Ok, here it is. Let us consider a 3D non singular potential that can model the confinement phenomenon. For example, a 3D oscillator potential V(
r)=mω
2r2/2. The Schrodinger equation has exact solutions in this case. The lowest energy is equal to (3/2)ω.
Let us consider the Green's function G(
r,t|
r',t'). Its spectral representation is known:
G(
r,t|
r',t') = ∑ψ
n(
r)ψ
n(
r')e
-iEn(t-t') (0)
I will consider a particular case of G(
r,-it|
0,0) that I denote as G(
r,τ). Its equation is the following (kind of a diffusion rather than a wave equation):
{∂/∂τ - ∇
2/2m + V(
r)}G(
r,τ) = δ(
r)δ(τ) (1)
G(
r,τ) does not oscillates but decays when τ increases.
Let us now introduce the Green's function of free motion equation G
0(
r,τ):
{∂/∂τ - ∇
2/2m }G
0(
r,τ) = δ(
r)δ(τ) (2)
It is equal to G
0(
r,τ) = (m/2πτ)
3/2e
-mr2/2τ
From equation (1) one can obtain G(
r,τ) by the perturbation theory where a perturbation is the non singular potential:
G(
r,τ) ≈ G
0(
r,τ){1 - u(
r) + w(
r) - ...} (3)
where u(
r) and w(
r) are some integrals.
These integrals are simplified if
r=0. Then u(0)=u=(ωτ/2)
2, w(0)=α⋅u
2, α=19/30 for the 3D oscillator. The functions u, w, and the coefficient α can be calculated exactly for many non singular potentials.
However the expansion (3), especially if truncated, is good only for small τ. At large times τ the series {1 - u + α⋅u
2 - ...} diverges as the highest degree of time in it.
In order to extrapolate G(
0,τ) from (3) to finite and large times, we can apply a non-linear series summation. It is kind of summation of divergent series. We can use a positive Padé approximant [0/2]:
{1 - u + α⋅u
2} ≈ 1/{1 + u + (1-α)⋅u
2} (4)
This approximation represents well the function even at large τ. But at large times the Green's function is also well approximated with only one spectral term - the lowest level contribution:
G(
0,τ) ≈ |ψ
0(0)|
2e
-E0⋅τ (5)
Taking the logarithmic derivative of G(
0,τ) with help of (4) at big times, one obtains an estimation of E
0. For the 3D oscillator we obtained E
0 ≈ (2.62/2)ω.
We used also a Sommerfeld approximation:
{1 - u + α⋅u
2} ≈ (1+A⋅u)
B, A=2α-1, B=1/(1-2α) (6)
and we obtained E
0 ≈ (3/2)ω.
Of course, one should not choose too large time τ for calculating the logarithmic derivative of G(
0,τ): any approximation gets worse when τ→∞. The calculation point should be finite: just sufficient to use the one spectral term approximation (5) of the sum (0).
It is a very brief explanation of how one can do it.
It was published in the Soviet Journal of Nuclear Physics in 1985, V. 41, N.2, under the title:
"On sum rules for non-singular potentials in quantum mechanics."
Bob_for_short.
