"""
This file provides Pauli matices.
"""
import sympy
from sympy.physics.wigner import wigner_3j
# ==================================================
"""
Pauli matrices
"""
def sm0():
return sympy.Matrix([[1, 0], [0, 1]])
def smx():
return sympy.Matrix([[0, 1], [1, 0]])
def smy():
return sympy.Matrix([[0, -sympy.I], [sympy.I, 0]])
def smz():
return sympy.Matrix([[1, 0], [0, -1]])
def ss0():
return sympy.symbols(r"\sigma_{0}", commutative=False)
def ssx():
return sympy.symbols(r"\sigma_{x}", commutative=False)
def ssy():
return sympy.symbols(r"\sigma_{y}", commutative=False)
def ssz():
return sympy.symbols(r"\sigma_{z}", commutative=False)
# ==================================================
[docs]
def pauli_matrix(i=4, matrix=True):
"""
Pauli matrices (σ0, σx, σy, σz)
Args:
i (int, optional): = 0(0), 1(x), 2(y), 3(z), other(all)
matrix (bool, optional): Matrix ?
Returns:
Matrix or sympy or dict: matrix or symbol or dict { str: Matrix }
"""
if matrix:
sm = [sm0(), smx(), smy(), smz()]
if i in (0, 1, 2, 3):
return sm[i]
else:
return {"0": sm[0], "x": sm[1], "y": sm[2], "z": sm[3]}
else:
ss = [ss0(), ssx(), ssy(), ssz()]
if i in (0, 1, 2, 3):
return ss[i]
else:
return {"0": ss[0], "x": ss[1], "y": ss[2], "z": ss[3]}
# ==================================================
[docs]
def pauli_matrix_n(s, n, s1, s2):
"""
Pauli matrices (σ00, σ11, σ10, σ1-1)
Args:
s (int): 0/1
n (int): 0/+1/-1
s1 (sympy): ±1/2
s2 (sympy): ±1/2
Returns:
sympy: matrix element
"""
if s == 0 and n == 0:
return 1 if s1 == s2 else 0
elif s == 1:
return sympy.Rational((-1) ** (s1 - 1 / 2)) * sympy.sqrt(6) * wigner_3j(1 / 2, 1 / 2, 1, -s1, s2, n)
else:
return 0
