Wikipedia comments: "In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of some physical system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger."
- h2
2m
* ∇2 Ψ + U Ψ = E Ψ
Do you have a Schrödinger equation to solve? Or any other math equations? For the next few months of 2014, we are willing to help you solve them using Calculus-level programming. To start, copy and modify the source code below in a file we'll call it {abc}{123}.fc where {abc} = your initials and {123} = any number or id; 8 characters max. for filename. Edit your {abc}{123}.fc file, especially lines starting with a "!" character.
E-mail us your {abc}{123}.fc file. We will compile & execute it and send you the output file. Try compiling you own code with our FC-Compiler program. Download and install FC-Compiler with more [~60] example problems to choose from. Learn how easy it works.
All "!" characters in columns 1 or 2 must be deleted before compiling. These "!" were added in order to point out areas needing work.
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User's Schrödinger Equation Source Code:
For 1-dimensional Schrödinger Equation use following:
global all
problem SchrodingerPDE ! time-independent version
C ------------------------------------------------------------------------
C --- Calculus Programming example: Schrödinger Equation; a PDE (1D) Initial
C --- Value Problem.
C ------------------------------------------------------------------------
C
C User parameters ...
h = 6.6260689633e-34 ! Js ... Planck's constant
C
C x-parameter initial settings: x ==> i
! xFinal=1: xPrint=xFinal/20
C
call xAxis !
end ! Stmt.s not necessary in IVP, but used in BVP version
model xAxis !
C ... Integrate over x-axis
C
x= 0: xPrt = xPrint: dx = xPrt / 10
! PSI = ???: PSIx = ???: PSIxx = ??? ! Initial Conditions @x=0
Initiate pegasus for PDE &
equations PSIxx/PSIx, PSIx/PSI of x step dx to xPrt
do while (x .lt. xFinal)
Integrate PDE by pegasus
if( x .ge. xPrt) then
print 79, x, PSI, PSIx, PSIxx
xPrt = xPrt + xPrint
end if
end do
79 format( 1h , f8.4, 1x, 3(g14.5, 1x))
end
model PDE ! Partial Differential Equation
! U = ???
! E = ??? ! energy of particle
PSIxx = 2*m/(h*h) * (E - U) * PSI
end
For 2-dimensional Schrödinger Equation use following:
global all
problem SchrodingerPDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: Schrödinger Equation; a PDE (2D) Initial
C --- Value Problem solved using Method of Lines.
C ------------------------------------------------------------------------
dynamic PSI, PSIx, PSIxx
C
C User parameters ...
h = 6.6260689633e-34 ! Js ... Planck's constant
! m = ... ! mass of the particle
! jpoints = 50 ! grid pts. over y-axis
C
C x-parameter initial settings: x ==> i
! xFinal = 1: xPrint = xFinal/20
C
C y-parameter initial settings: y ==> j
! yFinal = 1: dy = yFinal/(jpoints-1)
pi= 4*atan(1): jp= jpoints
allot PSI( jp), PSIx( jp), PSIxx( jp)
do 20 j = 1, jpoints ! Initial Conditions @y=0
PSI(j)= PSI0(j*dy): PSIx(j)= ???: PSIxx(j)= ???
20 continue
C
call xAxis !
end ! Stmt.s not necessary in IVP, but used in BVP version
model xAxis !
C ... Integrate over x-axis
C
x= 0: xPrt = xPrint: dx = xPrt / 10
Initiate athena for PDE &
equations PSIxx/PSIx, PSIx/PSI of x step dx to xPrt
do while (x .lt. xFinal)
Integrate PDE by athena
if( x .ge. xPrt) then
print 79, x, (PSI(jj), jj = 1, jp)
xPrt = xPrt + xPrint
end if
end do
79 format( 1h , f8.4, 1x, 10(g14.5, 1x))
end
model PDE ! Partial Differential Equation
C ! Method of Lines
do 40 jj = 2, jpoints - 1 ! System of ODEs
! U = ???
! E = ??? ! energy of particle
PSIyy = (PSI(jj+1) - 2*PSI(jj) + PSI(jj-1))/dy**2
PSIxx(jj)= 2*m/(h*h) * (E - U) * PSI - PSIyy
40 continue
! PSI(jp)= ???: PSIx(jp)= ???: PSIxx(jp)= ??? ! Boundary Conditions @x=L, if any
end
Fmodel PSI0(yy) ! Initial starting values @ x = 0
! if( yy .le. 0) then
! PSI0 = 0
! elseif( yy .lt. .5 ) then
! PSI0 = (1 - cos( 4 * pi * yy))/2
! else
! PSI0 = 0
! endif
end
