| from sympy.core import S, pi, Rational |
| from sympy.functions import assoc_laguerre, sqrt, exp, factorial, factorial2 |
|
|
|
|
| def 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 |
| |
| """ |
| n, l, nu, r = map(S, [n, l, nu, r]) |
|
|
| |
| n = n + 1 |
| C = sqrt( |
| ((2*nu)**(l + Rational(3, 2))*2**(n + l + 1)*factorial(n - 1))/ |
| (sqrt(pi)*(factorial2(2*n + 2*l - 1))) |
| ) |
| return C*r**(l)*exp(-nu*r**2)*assoc_laguerre(n - 1, l + S.Half, 2*nu*r**2) |
|
|
|
|
| def 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) |
| """ |
| return (2*n + l + Rational(3, 2))*hw |
|
|
