from __future__ import print_function, division
from sympy import symbols
from sympy.core import S, pi, Rational
from sympy.functions import sqrt, exp, factorial, gamma, tanh
from sympy.functions import assoc_laguerre as L
from sympy.functions import assoc_legendre as P
from sympy.physics.quantum.constants import hbar
from sympy import init_printing
init_printing()
The Morse potential is given by
$$ V(x) = D_e [1 - e^{-a(r-r_e)}])$$x, lam = symbols("x lambda")
n = symbols("n", integer=True)
def morse_psi_n(n, x, lam, xe):
Nn = sqrt((factorial(n)*(2*lam - 2*n - 10))/gamma(2*lam - n))
z = 2*lam*exp(-(x - xe))
psi = Nn*z**(lam - n -S(1)/2) * exp(-S(1)/2*z) * L(n, 2*lam - 2*n - 1, z)
return psi
def morse_E_n(n, lam):
return 1 - 1/lam**2*(lam - n - S(1)/2)**2
morse_E_n(0, lam)
from sympy.functions import cosh
The Pösch-Teller potential is given by
$$ V(x) = -\frac{\lambda(\lambda + 1)}{2} \operatorname{sech}^2(x)$$def posch_teller_psi_n(n, x, lam):
psi = P(lam, n, tanh(x))
return psi
posch_teller_psi_n(n, x, lam)
def morse_E_n(n, lam):
if n <= lam:
return -n**2/ 2
else:
raise ValueError("Lambda should not be greater than n.")
morse_E_n(5, 6)
Pöschl, G.; Teller, E. (1933). "Bemerkungen zur Quantenmechanik des anharmonischen Oszillators". Zeitschrift für Physik 83 (3–4): 143–151. doi:10.1007/BF01331132.
Siegfried Flügge Practical Quantum Mechanics (Springer, 1998)
Lekner, John (2007). "Reflectionless eigenstates of the sech2 potential". American Journal of Physics 875 (12): 1151–1157. doi:10.1119/1.2787015.
from IPython.core.display import HTML
def css_styling():
styles = open('./styles/custom_barba.css', 'r').read()
return HTML(styles)
css_styling()