def ang_mom_lowering(ang_mom=None, mag=None):
"""
Args:
ang_mom: The total angular momentum quantum number
mag: The magnetic quantum number
Returns:
If ang_mom == None and mag== None, the general formula for the lowering operator for spherical harmonics is returned.
Else, the formula for the lowering operator is computed using Dirac notation
Note:
Bra(str(j), str(","), str(m))*h_b*sqrt(j*(j+1)-m*(m-1))*Ket(str(j), str(','), str(m-1))
"""
L_x, L_y = symbols("L_x L_y")
if ang_mom == None and mag == None:
return Operator(L_x) - I * Operator(L_y)
if ang_mom == j and mag == m:
return Bra(str(j), str(","),
str(m)) * h_b * sqrt(j * (j + 1) - m * (m - 1)) * Ket(
str(j), str(','), str(m - 1))
if (str(ang_mom), str(","), str(mag)) == (str(ang_mom), str(','),
str(mag - 1)):
return h_b * sqrt(ang_mom * (ang_mom + 1) - mag * (mag - 1))
else:
return 0
def get_matrix_product_form(self):
self.ket_row = Matrix([Ket(s)
for s in self.state_components]).transpose()
self.bra_col = Matrix([Bra(s) for s in self.state_components])
self.form = UnevaluatedExpr(self.ket_row) * UnevaluatedExpr(
self.symmetrized_mat) * UnevaluatedExpr(self.bra_col)
return self.form
def ang_mom_raising(ang_mom=None, mag=None):
"""
Args:
ang_mom: The total angular momentum quantum number
mag: The magnetic quantum number
Returns:
If ang_mom == None and mag == None, the general formula for the raising operator for spherical harmonics is returned.
Else, the formula for the raising operator is computed using Dirac notation
Note:
Bra(str(j), str(","), str(m))*h_b*sqrt(j*(j+1)-m*(m+1))*Ket(str(j), str(','), str(m+1))
.. math::
\\langle j, m | \\hbar \\sqrt{ j (j+1) - m (m+1) } | j, m+1 \\rangle
"""
L_x, L_y, j, m = symbols("L_x L_y j m")
if ang_mom == None and mag == None:
return Operator(L_x) + I * Operator(L_y)
if ang_mom == j and mag == m:
return Bra(str(j), str(","),
str(m)) * h_b * sqrt(j * (j + 1) - m * (m + 1)) * Ket(
str(j), str(','), str(m + 1))
if (str(ang_mom), str(","), str(mag)) == (str(ang_mom), str(','),
str(mag + 1)):
return h_b * sqrt(ang_mom * (ang_mom + 1) - mag * (mag + 1))
else:
return 0
def simplify_ladder(expr):
"""
Args:
expr: the expression of interest
Returns:
The simplified expression, replacing any a_rasing(symbol)**2 with 0, a_lowering(symbol)**2 with 0, and
2*a_raising(symbol)*a_lowering(symbol) with (2*n+1).
Note:
The code does not "understand" the difference between a_raising(symbol)*a_lowering(symbol) and a_lowering(symbol)*a_raising(symbol)
which are mathematically different and produce different outcomes.
"""
symbol, n = symbols("symbol n")
return Bra(n) * (simplify(expand(simplify(expand(expr)))).replace(
(2 * a_raising(symbol) * a_lowering(symbol)),
(2 * n + 1)).replace(a_raising(symbol)**2, 0).replace(
a_lowering(symbol)**2, 0)) * Ket(n)
def a_lowering(A=None):
"""
Args:
A: this is usually either empty, (no parameter), "normalized" for the normalized ladder operator,
or a "symbol" for the symbolic representation of the parameter.
Returns:
Either the mathematic representation of the "a" lowering operator (commonly used in harmonic oscillator problems),
the normalized lowering operator (in dirac notation), or the symbol notation of the operator.
"""
m, omega, p, x, a_L, normalized, n, symbol = symbols(
"m omega p x a_- normalized n symbol")
if A == None:
return (1 / sqrt(2 * h_b * m * omega) * (I * p + m * omega * x))
if A == normalized:
return Bra(n) * sqrt(n) * Ket(n - 1)
if A == symbol:
return a_L
def ang_mom_2(ang_mom, mag):
"""
Args:
ang_mom: The total angular momentum quantum number
mag: The magnetic quantum number
Returns:
The L^2 vector magnitude eigenvalue for spherical harmonics.
Note:
Bra(str(j), str(","), str(m))*j*(j+1)*h_b**2*Ket(str(j), str(","), str(m))
"""
j, m = symbols("j, m")
if ang_mom == j and mag == m:
return Bra(str(j), str(","), str(m)) * j * (j + 1) * h_b**2 * Ket(
str(j), str(","), str(m))
if (str(ang_mom), str(","), str(mag)) == (str(ang_mom), str(","),
str(mag)):
return ang_mom * (ang_mom + 1) * h_b**2
else:
return 0
def ang_mom_z(ang_mom, mag):
"""
Args:
ang_mom: The total angular momentum quantum number
mag: The magnetic quantum number
Returns:
The L_z projection (in the z direction) eigenvalue for spherical harmonics.
Note:
Bra(str(j), str(","), str(m))*m*h_b*Ket(str(j), str(","), str(m))
"""
j, m = symbols("j, m")
if ang_mom == j and mag == m:
return Bra(str(j), str(","), str(m)) * m * h_b * Ket(
str(j), str(","), str(m))
if (str(ang_mom), str(","), str(mag)) == (str(ang_mom), str(","),
str(mag)):
return mag * h_b
else:
return 0
def __new__(cls, alpha):
return Ket.__new__(cls, alpha)
def __new__(cls, n):
return Ket.__new__(cls, n)
def __new__(cls, alpha):
return Ket.__new__(cls, alpha)
def __new__(cls, n):
return Ket.__new__(cls, n)
def __new__(cls, n):
if n not in [0, 1]:
raise ValueError("n must be 0 or 1")
return Ket.__new__(cls, n)
def __new__(cls, n):
if n not in [0, 1]:
raise ValueError("n must be 0 or 1")
return Ket.__new__(cls, n)
