# Wavefunctions

The fourth example creates wave-functions.

Wavefunctions.Quanty
-- We start with the same basis as before
NF=6
NB=0
IndexDn={0,2,4}
IndexUp={1,3,5}

-- We now can create wave functions. For this we
-- need three inputs:
-- (1) The number of Fermions and Bosons
-- (2) A list defining the wavefunctions. The lists
-- consists of a string defining a determinant by
-- its occupation: (1 occupied, 0 empty). One bit
-- for each fermion, 8 bit for a boson (i.e. a
-- boson can have an occupation between 0 and 255).
-- (3) A number giving the pre-factor of the
-- determinant.
psi0=NewWavefunction(NF, NB, {{"100000",math.sqrt(1/2)}, {"000001",math.sqrt(1/2)}})

-- We can print the function
print(psi0)

-- The name of the wave function is generic, we can
-- set it
psi0.Name = "psi 0"
print(psi0)

-- We now could create all one electron functions
-- in the basis of the p-shell.
psi0=NewWavefunction(NF,NB,{{"100000",1}})
psi0.Name = "psi(1 -1 1/2 -1/2)"
psi1=NewWavefunction(NF,NB,{{"001000",1}})
psi1.Name = "psi(1  0 1/2 -1/2)"
psi2=NewWavefunction(NF,NB,{{"000010",1}})
psi2.Name = "psi(1  1 1/2 -1/2)"
psi3=NewWavefunction(NF,NB,{{"010000",1}})
psi3.Name = "psi(1 -1 1/2  1/2)"
psi4=NewWavefunction(NF,NB,{{"000100",1}})
psi4.Name = "psi(1  0 1/2  1/2)"
psi5=NewWavefunction(NF,NB,{{"000001",1}})
psi5.Name = "psi(1  1 1/2  1/2)"

-- And print them
print("============================")
print(psi0)
print(psi1)
print(psi2)
print(psi3)
print(psi4)
print(psi5)
print("============================")

-- The wavefunctions should be orthonormal, we can
-- easily test this
psiList={psi0,psi1,psi2,psi3,psi4,psi5}
for i = 1, 6 do
for j = 1, 6 do
print("<",psiList[i].Name," | ", psiList[j].Name," > =",psiList[i] * psiList[j])
end
end
print("============================")
-- in table or matrix form:
psiList={psi0,psi1,psi2,psi3,psi4,psi5}
for i = 1, 6 do
for j = 1, 6 do
io.write(string.format("%3i ",psiList[i] * psiList[j]))
end
io.write("\n")
end

The output is:

Wavefunctions.out
WaveFunction: Wave Function
QComplex         =          0 (Real==0 or Complex==1)
N                =          2 (Number of basis functions used to discribe psi)
NFermionic modes =          6 (Number of fermions in the one particle basis)
NBosonic modes   =          0 (Number of bosons in the one particle basis)

#      pre-factor         Determinant
1   7.071067811865E-01       100000
2   7.071067811865E-01       000001

WaveFunction: psi 0
QComplex         =          0 (Real==0 or Complex==1)
N                =          2 (Number of basis functions used to discribe psi)
NFermionic modes =          6 (Number of fermions in the one particle basis)
NBosonic modes   =          0 (Number of bosons in the one particle basis)

#      pre-factor         Determinant
1   7.071067811865E-01       100000
2   7.071067811865E-01       000001

============================

WaveFunction: psi(1 -1 1/2 -1/2)
QComplex         =          0 (Real==0 or Complex==1)
N                =          1 (Number of basis functions used to discribe psi)
NFermionic modes =          6 (Number of fermions in the one particle basis)
NBosonic modes   =          0 (Number of bosons in the one particle basis)

#      pre-factor         Determinant
1   1.000000000000E+00       100000

WaveFunction: psi(1  0 1/2 -1/2)
QComplex         =          0 (Real==0 or Complex==1)
N                =          1 (Number of basis functions used to discribe psi)
NFermionic modes =          6 (Number of fermions in the one particle basis)
NBosonic modes   =          0 (Number of bosons in the one particle basis)

#      pre-factor         Determinant
1   1.000000000000E+00       001000

WaveFunction: psi(1  1 1/2 -1/2)
QComplex         =          0 (Real==0 or Complex==1)
N                =          1 (Number of basis functions used to discribe psi)
NFermionic modes =          6 (Number of fermions in the one particle basis)
NBosonic modes   =          0 (Number of bosons in the one particle basis)

#      pre-factor         Determinant
1   1.000000000000E+00       000010

WaveFunction: psi(1 -1 1/2  1/2)
QComplex         =          0 (Real==0 or Complex==1)
N                =          1 (Number of basis functions used to discribe psi)
NFermionic modes =          6 (Number of fermions in the one particle basis)
NBosonic modes   =          0 (Number of bosons in the one particle basis)

#      pre-factor         Determinant
1   1.000000000000E+00       010000

WaveFunction: psi(1  0 1/2  1/2)
QComplex         =          0 (Real==0 or Complex==1)
N                =          1 (Number of basis functions used to discribe psi)
NFermionic modes =          6 (Number of fermions in the one particle basis)
NBosonic modes   =          0 (Number of bosons in the one particle basis)

#      pre-factor         Determinant
1   1.000000000000E+00       000100

WaveFunction: psi(1  1 1/2  1/2)
QComplex         =          0 (Real==0 or Complex==1)
N                =          1 (Number of basis functions used to discribe psi)
NFermionic modes =          6 (Number of fermions in the one particle basis)
NBosonic modes   =          0 (Number of bosons in the one particle basis)

#      pre-factor         Determinant
1   1.000000000000E+00       000001

============================
<	psi(1 -1 1/2 -1/2)	 | 	psi(1 -1 1/2 -1/2)	 > =	1
<	psi(1 -1 1/2 -1/2)	 | 	psi(1  0 1/2 -1/2)	 > =	0
<	psi(1 -1 1/2 -1/2)	 | 	psi(1  1 1/2 -1/2)	 > =	0
<	psi(1 -1 1/2 -1/2)	 | 	psi(1 -1 1/2  1/2)	 > =	0
<	psi(1 -1 1/2 -1/2)	 | 	psi(1  0 1/2  1/2)	 > =	0
<	psi(1 -1 1/2 -1/2)	 | 	psi(1  1 1/2  1/2)	 > =	0
<	psi(1  0 1/2 -1/2)	 | 	psi(1 -1 1/2 -1/2)	 > =	0
<	psi(1  0 1/2 -1/2)	 | 	psi(1  0 1/2 -1/2)	 > =	1
<	psi(1  0 1/2 -1/2)	 | 	psi(1  1 1/2 -1/2)	 > =	0
<	psi(1  0 1/2 -1/2)	 | 	psi(1 -1 1/2  1/2)	 > =	0
<	psi(1  0 1/2 -1/2)	 | 	psi(1  0 1/2  1/2)	 > =	0
<	psi(1  0 1/2 -1/2)	 | 	psi(1  1 1/2  1/2)	 > =	0
<	psi(1  1 1/2 -1/2)	 | 	psi(1 -1 1/2 -1/2)	 > =	0
<	psi(1  1 1/2 -1/2)	 | 	psi(1  0 1/2 -1/2)	 > =	0
<	psi(1  1 1/2 -1/2)	 | 	psi(1  1 1/2 -1/2)	 > =	1
<	psi(1  1 1/2 -1/2)	 | 	psi(1 -1 1/2  1/2)	 > =	0
<	psi(1  1 1/2 -1/2)	 | 	psi(1  0 1/2  1/2)	 > =	0
<	psi(1  1 1/2 -1/2)	 | 	psi(1  1 1/2  1/2)	 > =	0
<	psi(1 -1 1/2  1/2)	 | 	psi(1 -1 1/2 -1/2)	 > =	0
<	psi(1 -1 1/2  1/2)	 | 	psi(1  0 1/2 -1/2)	 > =	0
<	psi(1 -1 1/2  1/2)	 | 	psi(1  1 1/2 -1/2)	 > =	0
<	psi(1 -1 1/2  1/2)	 | 	psi(1 -1 1/2  1/2)	 > =	1
<	psi(1 -1 1/2  1/2)	 | 	psi(1  0 1/2  1/2)	 > =	0
<	psi(1 -1 1/2  1/2)	 | 	psi(1  1 1/2  1/2)	 > =	0
<	psi(1  0 1/2  1/2)	 | 	psi(1 -1 1/2 -1/2)	 > =	0
<	psi(1  0 1/2  1/2)	 | 	psi(1  0 1/2 -1/2)	 > =	0
<	psi(1  0 1/2  1/2)	 | 	psi(1  1 1/2 -1/2)	 > =	0
<	psi(1  0 1/2  1/2)	 | 	psi(1 -1 1/2  1/2)	 > =	0
<	psi(1  0 1/2  1/2)	 | 	psi(1  0 1/2  1/2)	 > =	1
<	psi(1  0 1/2  1/2)	 | 	psi(1  1 1/2  1/2)	 > =	0
<	psi(1  1 1/2  1/2)	 | 	psi(1 -1 1/2 -1/2)	 > =	0
<	psi(1  1 1/2  1/2)	 | 	psi(1  0 1/2 -1/2)	 > =	0
<	psi(1  1 1/2  1/2)	 | 	psi(1  1 1/2 -1/2)	 > =	0
<	psi(1  1 1/2  1/2)	 | 	psi(1 -1 1/2  1/2)	 > =	0
<	psi(1  1 1/2  1/2)	 | 	psi(1  0 1/2  1/2)	 > =	0
<	psi(1  1 1/2  1/2)	 | 	psi(1  1 1/2  1/2)	 > =	1
============================
1   0   0   0   0   0
0   1   0   0   0   0
0   0   1   0   0   0
0   0   0   1   0   0
0   0   0   0   1   0
0   0   0   0   0   1