Source code for sympy.physics.qho_1d

from __future__ import print_function, division

from sympy.core import S, pi, Rational
from sympy.functions import hermite, sqrt, exp, factorial, Abs
from sympy.physics.quantum.constants import hbar

[docs]def psi_n(n, x, m, omega): """ Returns the wavefunction psi_{n} for the One-dimensional harmonic oscillator. ``n`` the "nodal" quantum number. Corresponds to the number of nodes in the wavefunction. n >= 0 ``x`` x coordinate ``m`` mass of the particle ``omega`` angular frequency of the oscillator Examples ======== >>> from sympy.physics.qho_1d import psi_n >>> from sympy import var >>> var("x m omega") (x, m, omega) >>> psi_n(0, x, m, omega) (m*omega)**(1/4)*exp(-m*omega*x**2/(2*hbar))/(hbar**(1/4)*pi**(1/4)) """ # sympify arguments n, x, m, omega = map(S, [n, x, m, omega]) nu = m * omega / hbar # normalization coefficient C = (nu/pi)**(S(1)/4) * sqrt(1/(2**n*factorial(n))) return C * exp(-nu* x**2 /2) * hermite(n, sqrt(nu)*x)
[docs]def E_n(n, omega): """ Returns the Energy of the One-dimensional harmonic oscillator ``n`` the "nodal" quantum number ``omega`` the harmonic oscillator angular frequency The unit of the returned value matches the unit of hw, since the energy is calculated as: E_n = hbar * omega*(n + 1/2) Examples ======== >>> from sympy.physics.qho_1d import E_n >>> from sympy import var >>> var("x omega") (x, omega) >>> E_n(x, omega) hbar*omega*(x + 1/2) """ return hbar * omega*(n + Rational(1, 2))
[docs]def coherent_state(n, alpha): """ Returns <n|alpha> for the coherent states of 1D harmonic oscillator. See ``n`` the "nodal" quantum number ``alpha`` the eigen value of annihilation operator """ return exp(- Abs(alpha)**2/2)*(alpha**n)/sqrt(factorial(n))