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documentation:language_reference:functions:rotate [2016/10/09 21:51] – created Maurits W. Haverkortdocumentation:language_reference:functions:rotate [2025/11/20 03:29] (current) – external edit 127.0.0.1
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 +====== 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 ====
 +<code Quanty 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)
 +</code>
 +
 +==== Result ====
 +<file Quanty_Output 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
 +</file>
 +
 +===== Table of contents =====
 +{{indexmenu>.#1}}

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