Quantum Harmonic Oscillator in 3-D
Quantum Harmonic Oscillator in 3-D¶
- sympy.physics.sho.E_nl(n, l, hw)[source]¶
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)[source]¶
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