pysces¶
-
pysces.PIB(x, L, n)[source]¶ - Parameters
x – a variable.
L – Length of the box.
n – an integer.
- Returns
The WaveFunction for Particle in a Box with respect to the chosen parameters.
-
pysces.PIB_2D(x, y, L_x, L_y, n)[source]¶ - Parameters
x – a variable.
y – a variable.
L_x – Length of the box in the x direction
L_y – Length of the box in the y direction
n – an integer.
- Returns
The WaveFunction for Particle in a Box in two dimensions with respect to the chosen parameters.
-
pysces.PIB_2D_normalized(x, y, L_x, L_y, n)[source]¶ - Parameters
x – a variable.
y – a variable.
L_x – Length of the box in the x direction
L_y – Length of the box in the y direction
n – an integer.
- Returns
The normalized WaveFunction for Particle in a Box, with respect to the chosen variables. This answer at this time cannot be calculated using the normalization_constant() function.
-
pysces.PIB_normalized(x, L, n)[source]¶ - Parameters
x – a variable.
L – Length of the box.
n – an integer.
- Returns
The normalized WaveFunction for Particle in a Box, with respect to the chosen variables. This answer can also be calculated using the normalization_constant() function.
-
pysces.a_lowering(A=None)[source]¶ - Parameters
A – this is usually either empty, (no parameter), “normalized” for the normalized ladder operator,
a "symbol" for the symbolic representation of the parameter. (or) –
- 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.
-
pysces.a_raising(A=None)[source]¶ - Parameters
A – this is usually either empty, (no parameter), “normalized” for the normalized ladder operator, or a “symbol” for the symbolic
of the parameter. (representation) –
- Returns
Either the mathematic representation of the “a” raising operator (commonly used in harmonic oscillator problems), the normalized raising operator (in dirac notation), or the symbol notation of the operator.
-
pysces.ang_mom_2(ang_mom, mag)[source]¶ - Parameters
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))
-
pysces.ang_mom_lowering(ang_mom=None, mag=None)[source]¶ - Parameters
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))
-
pysces.ang_mom_raising(ang_mom=None, mag=None)[source]¶ - Parameters
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))
\[\langle j, m | \hbar \sqrt{ j (j+1) - m (m+1) } | j, m+1 \rangle\]
-
pysces.ang_mom_x(ang_mom=None, mag=None)[source]¶ - Parameters
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 L_x operator for spherical harmonics is returned. Else, the formula for the L_x operator is computed using Dirac notation
-
pysces.ang_mom_z(ang_mom, mag)[source]¶ - Parameters
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))
-
pysces.comm_1(commutator_1, commutator_2, aux)[source]¶ This function is not used directly, it is only used in the comm() function below.
- Parameters
commutator_1 – The first operator in the commutator
commutator_2 – The second operator in the commutator
aux – The auxiliary function. This is defined below, as F(x).
- Returns
The commutator of commutator_1 and commutator_2 with respect to the auxiliary function.
-
pysces.comm_steps(commutator_1, commutator_2, aux)[source]¶ - Parameters
commutator_1 – the first operator
commutator_2 – the second operator
aux – the auxiliary function
- Returns
Three main steps used to solve a commutator. The first is the printed commutator in brackets, the second is the expansion, and the third output will be the answer. Please note that as of now, the code does not understand the addition/subtraction/multiplication/division of commutators, and so if there are multiple commutators in a line (ex: comm_steps(comm(A, B) - comm(C, D))) the code will not process this and therefore each individual commutator needs to have its own line.
-
pysces.conjugate(expr)[source]¶ - Parameters
x – The term of interest. This is commonly a WaveFunction.
- Returns
The complex conjugate of a WaveFunction. If there are imaginary terms, they get negated, and if there are no imaginary terms, the WaveFunction is not affected.
-
pysces.expectation_value(wavefunc_1, operator, wavefunc_2, lower, upper, var)[source]¶ Computes a symbolic expression for an expectation value of an operator operator with the two wavefunctions: <WaveFunc_1|operator|WaveFunc_2>
- Parameters
wavefunc_1 – The “bra” normalized WaveFunction
operator – The operator of interest
wavefunc_2 – The “ket” normalized WaveFunction
lower – The lower bound of the integral. If bounds are not listed, this is -oo
upper – The upper bound of the integral. If bounds are not listed, this is oo
var – What the integral is taken with respect to
- Returns
The expectation value for the given operator and normalized WaveFunction. An expectation value is the average value of an operator for a given WaveFunction.
-
pysces.expectation_value_steps(wavefunc_1, operator, wavefunc_2, lower, upper, var)[source]¶ Computes a symbolic expression for an expectation value of an operator operator with the two wavefunctions: <WaveFunc_1|operator|WaveFunc_2>
- Parameters
wavefunc_1 – The “bra” normalized WaveFunction
operator – The operator of interest
wavefunc_2 – The “ket” normalized WaveFunction
lower – The lower bound of the integral. If bounds are not listed, this is -oo
upper – The upper bound of the integral. If bounds are not listed, this is oo
var – What the integral is taken with respect to
- Returns
The steps on how to solve an expectation value problem. An expectation value is the average value of an operator for a given WaveFunction.
-
pysces.expression_replace(expr, var)[source]¶ This is only used within the “comm()” function.
expression_replace(expr, var)
- Parameters
expr – The expanded commutator to be replaced
var – The variable/parameter with respect to the chosen commutator.
- Returns
This replaces the Derivative(1, x) present in the expanded commutator with either Derivative(F(x), x) or Derivative(x*F(x), x).
Note
The above states “x”, but can be done for x, y, or z variables. The “expr” parameter is a str()
-
pysces.f(var)[source]¶ - Parameters
var – This is what the auxiliary function is with respect to. It should also match the parameters of the other arguments in the
function. (example (comm()) – if one operator is lin_mom(y), the auxiliary function should be f(y).
- Returns
This is the auxiliary function, commonly used in the comm() function. This simply returns “f(x)”, x being with chosen parameter.
-
pysces.factorization(expr, var)[source]¶ - Parameters
expr – the expression of interest
var – the variable of interest. This is most likely going to be the same variable used in the auxiliary function.
- Returns
The simplified commutator. This is only used when both of the operators in the commutator are angular momentum operators.
-
pysces.fixed_gaussian(alpha, x_0, p)[source]¶ - Parameters
alpha – alpha parameter of the gaussian
x_0 – x_0 parameter of the gaussian
p – p parameter of the gaussian
- Returns
The moving gaussian wave function.
-
pysces.hamiltonian(var)[source]¶ - Parameters
var – The variable for the given function. This is usually “x”, “y” or “z”.
- Returns
The Hamiltonian operator, made up of the kinetic energy operator and the general potential energy operator.
-
pysces.harmonic(condition=None)[source]¶ - Parameters
condition – the condition of the harmonic oscillator. Examples include: “ground_state” or “ladder”. This can also be left blank to give the general definition of a harmonic oscillator.
- Returns
The Harmonic Oscillator Hamiltonian. Note that “p” is the linear momentum operator, and so the first term is the kinetic_energy() function mentioned above.
-
pysces.kinetic_energy(var=None)[source]¶ - Parameters
var – The variable in which the derivative is with respect to.
- Returns
The kinetic energy operator, with respect to the chosen parameter.
Note
The “1” in the derivative is a placeholder, which will be replaced. If var == None, the general linear operator is printed, with all three positional arguments “x”, “y”, and “z”
-
pysces.kronecker(i, j)[source]¶ - Parameters
i – this is usually “i” and the first variable of the kronecker delta.
j – this is usually given a numerical value, and is the second variable of the kronecker delta.
- Returns
- Return type
This function does not print anything, but instead is used in conjunction with the code found in the kronecker_delta file on
-
pysces.laguerre(r, n)[source]¶ - Parameters
r – What the equation is with respect to. This is commonly “r” or “x”
n – The principle Quantum Number
- Returns
The Laguerre polynomial. This is commonly used to solve the laguerre_assoc() function.
-
pysces.laguerre_2(r, n)[source]¶ - Parameters
r – What the equation is with respect to. This is commonly “r” or “x”
n – The principle Quantum Number
- Returns
The Laguerre polynomial, without simplification and without the derivative computed.
-
pysces.laguerre_assoc(n, l)[source]¶ - Parameters
n – The principle Quantum Number
l – The angular momentum Quantum Number
- Returns
The Laguerre Associated polynomial, commonly used to solve the radial function.
-
pysces.lin_mom(var=None)[source]¶ - Parameters
var – The variable that the linear momentum operator is with respect to.
- Returns
The linear momentum operator, with respect to the parameter.
Note
The “1” in the derivative is a placeholder, which will be replaced. If var == None, the general linear operator is printed, with all three positional arguments “x”, “y”, and “z”
-
pysces.moving_gaussian(alpha, x_0, p)[source]¶ - Parameters
alpha – alpha parameter of the gaussian
x_0 – x_0 parameter of the gaussian
p – p parameter of the gaussian
- Returns
The moving gaussian wave function.
-
pysces.moving_gaussian_normalized(alpha, gamma, x_0, p_0)[source]¶ - Parameters
alpha – alpha parameter of the normalized gaussian
gamma – gamma parameter of the normalized gaussian
x_0 – x_0 parameter of the normalized gaussian
p_0 – p_0 parameter of the normalized gaussian
- Returns
The normalized gaussian WaveFunction, with respect to the chosen variables.
-
pysces.normalization_constant(wavefunc, lower, upper, var1, var2=None)[source]¶ - Parameters
wavefunc – The WaveFunction/expression of interest
lower – The lower bound of the integral. If bounds are not listed, this is -oo
upper – The upper bound of the integral. If bounds are not listed, this is oo
var1 – What the integral is taken with respect to
var2 – What the second integral is taken with respect to
- Returns
The normalization constant, with respect to the given parameters. To find the normalized WaveFunction, the output for normalization_constant() must be multiplied by the original WaveFunction. A normalized WaveFunction indicates that the probability of finding a particle within certain bounds must be equal to one.
Note
The double integration only works if they are integrated over the same bounds.
-
pysces.normalization_constant_steps(wavefunc, lower, upper, var1, var2=None)[source]¶ - Parameters
wavefunc – The WaveFunction/expression of interest
lower – The lower bound of the integral. If bounds are not listed, this is -oo
upper – The upper bound of the integral. If bounds are not listed, this is oo
var1 – What the (first) integral is taken with respect to
var2 – What the second integral is taken with respect to (if there are two integrals)
- Returns
The steps on how to solve a normalization problem.
-
pysces.overlap(WaveFunc_1, WaveFunc_2, lower, upper, var)[source]¶ Computes a symbolic expression for an overlap of two functions, <WaveFunc_1|WaveFunc_2>
- Parameters
WaveFunc_1 – The “bra” normalized WaveFunction
WaveFunc_2 – The “ket” normalized WaveFunction
var – What the integral is taken with respect to
lower – The lower bound of the integral. If bounds are not listed, this is -oo
upper – The upper bound of the integral. If bounds are not listed, this is oo
- Returns
The overlap of the two WaveFunctions of interest over given bounds.
-
pysces.planewave(x)[source]¶ - Parameters
x – What the function is with respect to.
- Returns
The PlaneWave function with respect to the parameter.
-
pysces.plot_function(func, lower1, upper1, B, lower2=None, upper2=None, C=None)[source]¶ - Parameters
func – The function/Normalized WaveFunction of interest
lower – The lower bound of the B domain (for Particle in a Box, 0)
upper – The upper bound of the B domain (for Particle in a Box, 1)
B – This is “x” usually (x-axis) - the domain
lower2 – The lower bound for the C range (for Particle in a Box, 0)
upper2 – The upper bound for the C range (for Particle in a Box, 1)
C – This is “y” usually (y-axis) - the range
- Returns
A plotted function of the function/Normalized WaveFunction of interest.
Note
The cell often has to be run TWICE in order to print the plot/graph.
-
pysces.potential_energy(var)[source]¶ - Parameters
var – The variable for the given function. This is usually “x”, “y” or “z”.
- Returns
The general potential energy operator, v(x)
-
pysces.simplify_ladder(expr)[source]¶ - Parameters
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.