Quantum Harmonic Oscillator in 3-D

sympy.physics.sho.E_nl(n, l, hw)

Returns the Energy of an isotropic harmonic oscillator.

Parameters:

n :

The “nodal” quantum number.

l :

The orbital angular momentum.

hw :

The harmonic oscillator parameter.

Notes

The unit of the returned value matches the unit of hw, since the energy is calculated as:

E_nl = (2*n + l + 3/2)*hw

Examples

>>> from sympy.physics.sho import E_nl
>>> from sympy import symbols
>>> x, y, z = symbols('x, y, z')
>>> E_nl(x, y, z)
z*(2*x + y + 3/2)

sympy.physics.sho.R_nl(n, l, nu, r)

Returns the radial wavefunction R_{nl} for a 3d isotropic harmonic oscillator.

Parameters:

n :

The “nodal” quantum number. Corresponds to the number of nodes in the wavefunction. n >= 0

l :

The quantum number for orbital angular momentum.

nu :

mass-scaled frequency: nu = m*omega/(2*hbar) where \(m\) is the mass and \(omega\) the frequency of the oscillator. (in atomic units nu == omega/2)

r :

Radial coordinate.

Examples

>>> from sympy.physics.sho import R_nl
>>> from sympy.abc import r, nu, l
>>> R_nl(0, 0, 1, r)
2*2**(3/4)*exp(-r**2)/pi**(1/4)
>>> R_nl(1, 0, 1, r)
4*2**(1/4)*sqrt(3)*(3/2 - 2*r**2)*exp(-r**2)/(3*pi**(1/4))

l, nu and r may be symbolic:

>>> R_nl(0, 0, nu, r)
2*2**(3/4)*sqrt(nu**(3/2))*exp(-nu*r**2)/pi**(1/4)
>>> R_nl(0, l, 1, r)
r**l*sqrt(2**(l + 3/2)*2**(l + 2)/factorial2(2*l + 1))*exp(-r**2)/pi**(1/4)

The normalization of the radial wavefunction is:

>>> from sympy import Integral, oo
>>> Integral(R_nl(0, 0, 1, r)**2*r**2, (r, 0, oo)).n()
1.00000000000000
>>> Integral(R_nl(1, 0, 1, r)**2*r**2, (r, 0, oo)).n()
1.00000000000000
>>> Integral(R_nl(1, 1, 1, r)**2*r**2, (r, 0, oo)).n()
1.00000000000000