..\DiffEqORMAT
Introduction
In this document the harmonic oscillator will be used as a test case for the ability to approximate the wave function by a complex form similar to the WKB form. The beginning of this is a visual hand fit of a harmonic oscillator potential to the bottom of the resonance potential.
Figure 1 plotv1 is the resonance potential. plotv is a Harmonic oscillator that approximates the resonance potential.
X'S FOR E=POT(X) -4.3250000000000002
-0.8496377147287321
0.8496377147287320
Potential.for
FUNCTION V(X,dVdx,IPOT)
DATA A,W/5D0,3D0/
SELECT CASE(IPOT)
…
CASE(4)
V=(X/(.45d0*W))**2-4.8168436111126578D0
DVDX=2*X/(.45d0*W)**2
…
END SELECT
RETURN
END
Start at the potential minimum
1)
Define
2)
3)
The Schrodinger equation is
4)
Or
5)
So that for the ground state
6)
AlphaFind.for
ALPHA,EH 0.5237828008789240 -4.2930608102337340
EH0=-3.2454952084758855 1ST excited H.O. state.
The excited states are defined in ..\Raising.docx
7)
8)
9)
The file HOTheory.out is found by HOBli.wpj. I used 65 points so that the difference between the differential equation and the simple theory is easy to see.
EulerExt.wpj
EulerExt.for
Case(1)
10)
...
CASE(1) ! Normal Harmonic Oscillator Ground State exp(-alpha*x2)
…
P0=-alpha*x0**2 !V starts at 0
Z0=-2*alpha*x0
…
CALL ZFIND(NAZ,IPOT,EH,X0,Z0,XEND)
…
CALL zToP(NAZ,Nap,p0)
…
CALL pToPsi(NAp,NAPsi,ISELECT)
Figure 2 z(x) as found by EulerExt
The function z(x) = -2αx was expected to follow the straight line indefinitely. The eigenvalue with 3000 steps across the region shown is exact to the 9 digits printed out for all values of x.
Figure 3dz/dx as a function of x
Figure 4 the wave function from z (exp(p(x)) comparedwith the theoretical H.O. ground state
Figure 5EulerExt using f t t versus theory for Harmonic oscillator
Figure 6 Data from figure above on log scale.
Case(2) WKB inspired
11)
CASE(2) !H.O.WKB starting
c p=-isqrt(EH-V(x0))*x WKB
c z=dp/dx=-i*sqrt(EH-V(x0))+i*x*dVdx0/sqrt(EH-V(x0))
…
V0=V(x0,dVdx0,IPOT)
if(EH.GT.V0)then
srt=sqrt(2*(EH-V0))
p0=-ci*srt*x0
z0=-ci*(srt-x0*dVdx0/srt)
else
srt=sqrt(2*(V0-EH))
p0=srt*x0
z0=srt+x0*dVdx0/srt
endif
…
CALL ZFIND(NAZ,IPOT,EH,X0,Z0,XEND)
…
CALL zToP(NAZ,Nap,p0)
…
CALL pToPsi(NAp,NAPsi,ISELECT)
Figure 7 zcase2 real and imaginary and zcase1 for comparison
The integral of the imaginary part of z is π/2. The term exp(iπ/2) is a complex way of writing -1. This means that for large values of x, the wave function goes to an incorrect -∞, rather than the incorrect +∞ seen in figure 6.
Figure 8 zcase2 1 4 is real(dzdx): zcase2 1 5 is imag(dzdx)
The energy is the ground state energy, -4.29306081, for all values of x.
Figure 9PsiCase2 1 3 = imag(PsiCase2)
Case(3) WKB inspired
12)
Only the two initial lines are changed from case(2)
Figure 10 zcase3 real(z); zcase3 1 3 imag(z); zcase1 for comparison.
Figure 11 Psicase3 1 3 = imag(Psi)
HO 1stexcited state
At x=0
2ND excited stateORMAT
1)
This state contains the ground state in chat which makes it harder to minimize the variance. – it keeps wanting to go to the ground state.
So that in one dimension