Sample Problems > Applications > Quantun_Mechanics > morse

{ MORSE_POTENTIAL.PDE

Submitted by Ali Reza Ghaffari, 07/21/2016.

This is a Quantum Mechanic example that shows the power of Flexpde to solve such examples.

We want to solve the Schrodinger equation for Morse Potential V(x)=V0(1-exp(-alpha*x))^2 and

find the Eigen values and functions. The exact energies can be extracted from the formula below.

E[n] := h*2^(1/2)*(V0*alpha^2/m0)^(1/2)*(n+1/2)-1/2*alpha^2*h^2/m0*(n+1/2)^2

For n=0 to 4 :

E[0] := 3.037277660

E[1] := 8.361832980

E[2] := 12.68638830

E[3] := 16.01094362

E[4] := 18.33549894

You can compare the results of this script with above energies.

}

TITLE 'Morse Potential'

COORDINATES CARTESIAN1

VARIABLES Phi

SELECT

modes=6

NGRID=30

ERRLIM=1e-3

DEFINITIONS

volt

hbar=1

m0=1

v0=20

a=10 ! the renge of integrals

alpha=1

volt=v0*(1-exp(-alpha*x))^2

N=integral(phi^2)

EQUATIONS

Phi : (-hbar^2/2/m0)*(dx(dx(Phi)))+volt*Phi-LAMBDA*Phi=0

BOUNDARIES

REGION 1

START (-3*a) point value(phi)=0

LINE TO (3*a) point value(phi)=0

!MONITORS

! no monitors since problem solves so fast

PLOTS

ELEVATION(Phi+lambda,volt) FROM (-1) to (6) zoom (-1,0,6,10)

SUMMARY

REPORT(LAMBDA)

END

morse_potential