Quantum Harmonic Oscillator in 3-D¶
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sympy.physics.sho.
E_nl
(n, l, hw)[source]¶ Returns the Energy of an isotropic harmonic oscillator
n
- the “nodal” quantum number
l
- the orbital angular momentum
hw
- the harmonic oscillator parameter.
The unit of the returned value matches the unit of hw, since the energy is calculated as:
E_nl = (2*n + l + 3/2)*hwExamples
>>> 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)
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sympy.physics.sho.
R_nl
(n, l, nu, r)[source]¶ Returns the radial wavefunction R_{nl} for a 3d isotropic harmonic oscillator.
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 import var >>> var("r nu l") (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)*(-2*r**2 + 3/2)*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