Rotate
Rotate($M$, $R$), Rotate($\psi$, $R$), Rotate($O$, $R$) or Rotate($TB$, $R$) rotates the basis of a quadratic matrix, a wave-function, an operator, or a tight-binding object (passive transformation). For a matrix it thus returns $R^\ast \cdot M \cdot R^T$, where $R^\ast$ denotes the complex conjugate and $R^T$ the transpose of the rotation matrix $R$.
$R$ needs to be a matrix of dimension $N_1\times N_2$, where $N_2$ equals the number of rows of $M$, or $N_{Fermion}+N_{Boson}$ of $\psi$, $O$ or $TB$. The rotation matrix $R$ is not required to be quadratic, it is therefore possible to use rotations to change the number of dimensions of the Hilbert-space.
$R^\ast \cdot R^T = 1$ is not checked by Quanty, to allow scaling rotations.
Input
-
$M$, $\psi$, $O$ or $TB$ : a quadratic (complex or real valued) matrix, a wave-function, an operator, or a tight-binding object.
-
$R$ : a complex or real valued generalised rotation matrix.
Output
-
$M^\prime$, $\psi^\prime$, $O^\prime$ or $TB^\prime$ : the rotated quadratic matrix, wave-function, operator, or tight-binding object.
Example
A small example:
Input
- Rotate.Quanty
dofile("../definitions.Quanty") OpppxR = Chop(Rotate(Opppx,rotmat)) OpppyR = Chop(Rotate(Opppy,rotmat)) OpppzR = Chop(Rotate(Opppz,rotmat)) print("The matrix rotating an operator on the basis of spherical harmonics with alternating spin to a basis of kubic harmonics with alternating spin for a p orbital is:") print(rotmat) print("The operator adding a potential to the px orbital on a basis of spherical harmonics is: ",Opppx) print("The operator adding a potential to the px orbital on a basis of kubic harmonics is: ",OpppxR) print("The operator adding a potential to the py orbital on a basis of spherical harmonics is: ",Opppy) print("The operator adding a potential to the py orbital on a basis of kubic harmonics is: ",OpppyR) print("The operator adding a potential to the pz orbital on a basis of spherical harmonics is: ",Opppz) print("The operator adding a potential to the pz orbital on a basis of kubic harmonics is: ",OpppzR) psixdnR = Chop(Rotate(psixdn,rotmat)) psixupR = Chop(Rotate(psixup,rotmat)) psiydnR = Chop(Rotate(psiydn,rotmat)) psiyupR = Chop(Rotate(psiyup,rotmat)) psizdnR = Chop(Rotate(psizdn,rotmat)) psizupR = Chop(Rotate(psizup,rotmat)) print("The px orbital with spin down on a basis of spherical harmonics is: ",psixdn) print("The px orbital with spin down on a basis of kubic harmonics is: ",psixdnR) print("The py orbital with spin down on a basis of spherical harmonics is: ",psiydn) print("The py orbital with spin down on a basis of kubic harmonics is: ",psiydnR) print("The pz orbital with spin down on a basis of spherical harmonics is: ",psizdn) print("The pz orbital with spin down on a basis of kubic harmonics is: ",psizdnR) print("Counting the px orbital on a basis of spherical harmonics: ",psixdn *Opppx *psixdn) print("Counting the px orbital on a basis of kubic harmonics: ",psixdnR*OpppxR*psixdnR) print("Counting the py orbital on a basis of spherical harmonics: ",psiydn *Opppy *psiydn) print("Counting the py orbital on a basis of kubic harmonics: ",psiydnR*OpppyR*psiydnR) print("Counting the pz orbital on a basis of spherical harmonics: ",psizdn *Opppz *psizdn) print("Counting the pz orbital on a basis of kubic harmonics: ",psizdnR*OpppzR*psizdnR)
Result
- Rotate.out
The matrix rotating an operator on the basis of spherical harmonics with alternating spin to a basis of kubic harmonics with alternating spin for a p orbital is: { { 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 , 0 } , { 0 , 0.70710678118655 , 0 , 0 , 0 , -0.70710678118655 } , { (0 + 0.70710678118655 I) , 0 , 0 , 0 , (0 + 0.70710678118655 I) , 0 } , { 0 , (0 + 0.70710678118655 I) , 0 , 0 , 0 , (0 + 0.70710678118655 I) } , { 0 , 0 , 1 , 0 , 0 , 0 } , { 0 , 0 , 0 , 1 , 0 , 0 } } The operator adding a potential to the px orbital on a basis of spherical harmonics is: Operator: Opp px QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 2 (largest number of product of lader operators) NFermionic modes = 6 (Number of fermionic modes (site, spin, orbital, ...) in the one particle basis) NBosonic modes = 0 (Number of bosonic modes (phonon modes, ...) in the one particle basis) Operator of Length 2 QComplex = 0 (Real==0 or Complex==1) N = 8 (number of operators of length 2) C 0 A 0 | 4.999999999999999E-01 C 1 A 1 | 4.999999999999999E-01 C 4 A 4 | 4.999999999999999E-01 C 5 A 5 | 4.999999999999999E-01 C 4 A 0 | -5.000000000000000E-01 C 5 A 1 | -5.000000000000000E-01 C 0 A 4 | -5.000000000000000E-01 C 1 A 5 | -5.000000000000000E-01 The operator adding a potential to the px orbital on a basis of kubic harmonics is: Operator: Opp px QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 2 (largest number of product of lader operators) NFermionic modes = 6 (Number of fermionic modes (site, spin, orbital, ...) in the one particle basis) NBosonic modes = 0 (Number of bosonic modes (phonon modes, ...) in the one particle basis) Operator of Length 2 QComplex = 0 (Real==0 or Complex==1) N = 2 (number of operators of length 2) C 0 A 0 | 1.000000000000000E+00 C 1 A 1 | 1.000000000000000E+00 The operator adding a potential to the py orbital on a basis of spherical harmonics is: Operator: Opp py QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 2 (largest number of product of lader operators) NFermionic modes = 6 (Number of fermionic modes (site, spin, orbital, ...) in the one particle basis) NBosonic modes = 0 (Number of bosonic modes (phonon modes, ...) in the one particle basis) Operator of Length 2 QComplex = 0 (Real==0 or Complex==1) N = 8 (number of operators of length 2) C 0 A 0 | 4.999999999999999E-01 C 1 A 1 | 4.999999999999999E-01 C 4 A 4 | 4.999999999999999E-01 C 5 A 5 | 4.999999999999999E-01 C 4 A 0 | 5.000000000000000E-01 C 5 A 1 | 5.000000000000000E-01 C 0 A 4 | 5.000000000000000E-01 C 1 A 5 | 5.000000000000000E-01 The operator adding a potential to the py orbital on a basis of kubic harmonics is: Operator: Opp py QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 2 (largest number of product of lader operators) NFermionic modes = 6 (Number of fermionic modes (site, spin, orbital, ...) in the one particle basis) NBosonic modes = 0 (Number of bosonic modes (phonon modes, ...) in the one particle basis) Operator of Length 2 QComplex = 0 (Real==0 or Complex==1) N = 2 (number of operators of length 2) C 2 A 2 | 1.000000000000000E+00 C 3 A 3 | 1.000000000000000E+00 The operator adding a potential to the pz orbital on a basis of spherical harmonics is: Operator: Opp pz QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 2 (largest number of product of lader operators) NFermionic modes = 6 (Number of fermionic modes (site, spin, orbital, ...) in the one particle basis) NBosonic modes = 0 (Number of bosonic modes (phonon modes, ...) in the one particle basis) Operator of Length 2 QComplex = 0 (Real==0 or Complex==1) N = 2 (number of operators of length 2) C 2 A 2 | 9.999999999999999E-01 C 3 A 3 | 9.999999999999999E-01 The operator adding a potential to the pz orbital on a basis of kubic harmonics is: Operator: Opp pz QComplex = 0 (Real==0 or Complex==1 or Mixed==2) MaxLength = 2 (largest number of product of lader operators) NFermionic modes = 6 (Number of fermionic modes (site, spin, orbital, ...) in the one particle basis) NBosonic modes = 0 (Number of bosonic modes (phonon modes, ...) in the one particle basis) Operator of Length 2 QComplex = 0 (Real==0 or Complex==1) N = 2 (number of operators of length 2) C 4 A 4 | 9.999999999999999E-01 C 5 A 5 | 9.999999999999999E-01 The px orbital with spin down on a basis of spherical harmonics is: 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 000010 The px orbital with spin down on a basis of kubic harmonics is: WaveFunction: Wave Function 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 The py orbital with spin down on a basis of spherical harmonics is: WaveFunction: Wave Function QComplex = 1 (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 +I pre-factor Determinant 1 0.000000000000E+00 7.071067811865E-01 100000 2 0.000000000000E+00 7.071067811865E-01 000010 The py orbital with spin down on a basis of kubic harmonics is: WaveFunction: Wave Function 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 The pz orbital with spin down on a basis of spherical harmonics is: WaveFunction: Wave Function 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 The pz orbital with spin down on a basis of kubic harmonics is: WaveFunction: Wave Function 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 Counting the px orbital on a basis of spherical harmonics: 1 Counting the px orbital on a basis of kubic harmonics: 1 Counting the py orbital on a basis of spherical harmonics: 1 Counting the py orbital on a basis of kubic harmonics: 1 Counting the pz orbital on a basis of spherical harmonics: 1 Counting the pz orbital on a basis of kubic harmonics: 1
