Hydrogen Wavefunctions

sympy.physics.hydrogen.E_nl(n, Z=1)

Returns the energy of the state (n, l) in Hartree atomic units.

The energy does not depend on “l”.

Parameters:

n : integer

Principal Quantum Number which is an integer with possible values as 1, 2, 3, 4,…

Z :

Atomic number (1 for Hydrogen, 2 for Helium, …)

Examples

>>> from sympy.physics.hydrogen import E_nl
>>> from sympy.abc import n, Z
>>> E_nl(n, Z)
-Z**2/(2*n**2)
>>> E_nl(1)
-1/2
>>> E_nl(2)
-1/8
>>> E_nl(3)
-1/18
>>> E_nl(3, 47)
-2209/18

sympy.physics.hydrogen.E_nl_dirac(

n,

l,

spin_up=True,

Z=1,

c=137.035999037000,

)

Returns the relativistic energy of the state (n, l, spin) in Hartree atomic units.

The energy is calculated from the Dirac equation. The rest mass energy is not included.

Parameters:

n : integer

Principal Quantum Number which is an integer with possible values as 1, 2, 3, 4,…

l : integer

l is the Angular Momentum Quantum Number with values ranging from 0 to n-1.

spin_up :

True if the electron spin is up (default), otherwise down

Z :

Atomic number (1 for Hydrogen, 2 for Helium, …)

c :

Speed of light in atomic units. Default value is 137.035999037, taken from https://arxiv.org/abs/1012.3627

Examples

>>> from sympy.physics.hydrogen import E_nl_dirac
>>> E_nl_dirac(1, 0)
-0.500006656595360

>>> E_nl_dirac(2, 0)
-0.125002080189006
>>> E_nl_dirac(2, 1)
-0.125000416028342
>>> E_nl_dirac(2, 1, False)
-0.125002080189006

>>> E_nl_dirac(3, 0)
-0.0555562951740285
>>> E_nl_dirac(3, 1)
-0.0555558020932949
>>> E_nl_dirac(3, 1, False)
-0.0555562951740285
>>> E_nl_dirac(3, 2)
-0.0555556377366884
>>> E_nl_dirac(3, 2, False)
-0.0555558020932949

sympy.physics.hydrogen.Psi_nlm(n, l, m, r, phi, theta, Z=1)

Returns the Hydrogen wave function psi_{nlm}. It’s the product of the radial wavefunction R_{nl} and the spherical harmonic Y_{l}^{m}.

Parameters:

n : integer

Principal Quantum Number which is an integer with possible values as 1, 2, 3, 4,…

l : integer

l is the Angular Momentum Quantum Number with values ranging from 0 to n-1.

m : integer

m is the Magnetic Quantum Number with values ranging from -l to l.

r :

radial coordinate

phi :

azimuthal angle

theta :

polar angle

Z :

atomic number (1 for Hydrogen, 2 for Helium, …)

Everything is in Hartree atomic units.

Examples

>>> from sympy.physics.hydrogen import Psi_nlm
>>> from sympy import Symbol
>>> r=Symbol("r", positive=True)
>>> phi=Symbol("phi", real=True)
>>> theta=Symbol("theta", real=True)
>>> Z=Symbol("Z", positive=True, integer=True, nonzero=True)
>>> Psi_nlm(1,0,0,r,phi,theta,Z)
Z**(3/2)*exp(-Z*r)/sqrt(pi)
>>> Psi_nlm(2,1,1,r,phi,theta,Z)
-Z**(5/2)*r*exp(I*phi)*exp(-Z*r/2)*sin(theta)/(8*sqrt(pi))

Integrating the absolute square of a hydrogen wavefunction psi_{nlm} over the whole space leads 1.

The normalization of the hydrogen wavefunctions Psi_nlm is:

>>> from sympy import integrate, conjugate, pi, oo, sin
>>> wf=Psi_nlm(2,1,1,r,phi,theta,Z)
>>> abs_sqrd=wf*conjugate(wf)
>>> jacobi=r**2*sin(theta)
>>> integrate(abs_sqrd*jacobi, (r,0,oo), (phi,0,2*pi), (theta,0,pi))
1

sympy.physics.hydrogen.R_nl(n, l, r, Z=1)

Returns the Hydrogen radial wavefunction R_{nl}.

Parameters:

n : integer

Principal Quantum Number which is an integer with possible values as 1, 2, 3, 4,…

l : integer

l is the Angular Momentum Quantum Number with values ranging from 0 to n-1.

r :

Radial coordinate.

Z :

Atomic number (1 for Hydrogen, 2 for Helium, …)

Everything is in Hartree atomic units.

Examples

>>> from sympy.physics.hydrogen import R_nl
>>> from sympy.abc import r, Z
>>> R_nl(1, 0, r, Z)
2*sqrt(Z**3)*exp(-Z*r)
>>> R_nl(2, 0, r, Z)
sqrt(2)*(-Z*r + 2)*sqrt(Z**3)*exp(-Z*r/2)/4
>>> R_nl(2, 1, r, Z)
sqrt(6)*Z*r*sqrt(Z**3)*exp(-Z*r/2)/12

For Hydrogen atom, you can just use the default value of Z=1:

>>> R_nl(1, 0, r)
2*exp(-r)
>>> R_nl(2, 0, r)
sqrt(2)*(2 - r)*exp(-r/2)/4
>>> R_nl(3, 0, r)
2*sqrt(3)*(2*r**2/9 - 2*r + 3)*exp(-r/3)/27

For Silver atom, you would use Z=47:

>>> R_nl(1, 0, r, Z=47)
94*sqrt(47)*exp(-47*r)
>>> R_nl(2, 0, r, Z=47)
47*sqrt(94)*(2 - 47*r)*exp(-47*r/2)/4
>>> R_nl(3, 0, r, Z=47)
94*sqrt(141)*(4418*r**2/9 - 94*r + 3)*exp(-47*r/3)/27

The normalization of the radial wavefunction is:

>>> from sympy import integrate, oo
>>> integrate(R_nl(1, 0, r)**2 * r**2, (r, 0, oo))
1
>>> integrate(R_nl(2, 0, r)**2 * r**2, (r, 0, oo))
1
>>> integrate(R_nl(2, 1, r)**2 * r**2, (r, 0, oo))
1

It holds for any atomic number:

>>> integrate(R_nl(1, 0, r, Z=2)**2 * r**2, (r, 0, oo))
1
>>> integrate(R_nl(2, 0, r, Z=3)**2 * r**2, (r, 0, oo))
1
>>> integrate(R_nl(2, 1, r, Z=4)**2 * r**2, (r, 0, oo))
1

sympy.physics.hydrogen.Y_lm(l, m, phi, theta)

Returns the spherical harmonic or angular function Y_{l}^{m}.

Parameters:

l : integer

l is the Angular Momentum Quantum Number with values ranging from 0 to n-1.

m : iteger

m is the Magnetic Quantum Number with values ranging from -l to l.

phi :

azimuthal angle

theta :

polar angle

Notes

This function follows the Condon-Shortley phase convention and may return real or complex values depending on values for l and m.

Examples

>>> import sympy
>>> from sympy.physics.hydrogen import Y_lm
>>> phi, theta = sympy.symbols('phi theta', real=True)
>>> Y_lm(1, 0, phi, theta)
sqrt(3)*cos(theta)/(2*sqrt(pi))
>>> Y_lm(2, -1, phi, theta).simplify()
sqrt(30)*exp(-I*phi)*sin(2*theta)/(8*sqrt(pi))
>>> Y_lm(1, -1, phi, theta).simplify()
sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi))
>>> Y_lm(1, 1, 0, 1).evalf()
-0.290723302201011

Find angular nodes in dz2 atomic orbital by solving for zero probability. This returns the node theta angles in radians.

>>> dz2 = Y_lm(2, 0, phi, theta)**2
>>> [sol.evalf() for sol in sympy.solve(dz2)]
[4.0969092717143, 5.32786868905508, 2.18627603546528, 0.955316618124509]

See also

Z_lm, sympy.functions.special.spherical_harmonics.Ynm, sympy.functions.special.spherical_harmonics.Ynm_c, sympy.functions.special.spherical_harmonics.Znm

References

[R751]

https://en.wikipedia.org/wiki/Spherical_harmonics

[R752]

https://en.wikipedia.org/wiki/Table_of_spherical_harmonics

[R753]

https://mathworld.wolfram.com/SphericalHarmonic.html

[R754]

https://functions.wolfram.com/Polynomials/SphericalHarmonicY/

[R755]

https://dlmf.nist.gov/14.30

sympy.physics.hydrogen.Z_lm(l, m, phi, theta)

Returns the real spherical harmonic or angular function Y_{l}^{m}.

Parameters:

l : integer

l is the Angular Momentum Quantum Number with values ranging from 0 to n-1.

m : iteger

m is the Magnetic Quantum Number with values ranging from -l to l.

phi :

azimuthal angle

theta :

polar angle

Notes

This function follows the Condon-Shortley phase convention.

If the imaginary components do not cancel, try setting your phi and theta symbol assumptions to real. See below.

Examples

>>> import sympy
>>> from sympy.physics.hydrogen import Z_lm
>>> phi, theta = sympy.symbols('phi theta', real=True)
>>> Z_lm(1, 0, phi, theta)
sqrt(3)*cos(theta)/(2*sqrt(pi))
>>> Z_lm(2, -1, phi, theta).simplify()
-sqrt(15)*sin(phi)*sin(2*theta)/(4*sqrt(pi))
>>> Z_lm(1, -1, phi, theta).simplify()
-sqrt(3)*sin(phi)*sin(theta)/(2*sqrt(pi))
>>> Z_lm(1, 1, 0, 1).evalf()
-0.411144836870562

Find angular nodes in dz2 atomic orbital by solving for zero probability. This returns the node theta angles in radians.

>>> dz2 = Z_lm(2, 0, phi, theta)**2
>>> [sol.evalf() for sol in sympy.solve(dz2)]
[4.0969092717143, 5.32786868905508, 2.18627603546528, 0.955316618124509]

See also

Y_lm, sympy.functions.special.spherical_harmonics.Ynm, sympy.functions.special.spherical_harmonics.Ynm_c, sympy.functions.special.spherical_harmonics.Znm

References

[R756]

https://en.wikipedia.org/wiki/Spherical_harmonics

[R757]

https://en.wikipedia.org/wiki/Table_of_spherical_harmonics

[R758]

https://mathworld.wolfram.com/SphericalHarmonic.html

[R759]

https://functions.wolfram.com/Polynomials/SphericalHarmonicY/

[R760]

https://dlmf.nist.gov/14.30