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from sympy.core import S, pi, Rational
from sympy.functions import hermite, sqrt, exp, factorial, Abs
from sympy.physics.quantum.constants import hbar
def psi_n(n, x, m, omega):
"""
Returns the wavefunction psi_{n} for the One-dimensional harmonic oscillator.
Parameters
==========
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.abc import m, x, 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)**Rational(1, 4) * sqrt(1/(2**n*factorial(n)))
return C * exp(-nu* x**2 /2) * hermite(n, sqrt(nu)*x)
def E_n(n, omega):
"""
Returns the Energy of the One-dimensional harmonic oscillator.
Parameters
==========
n :
The "nodal" quantum number.
omega :
The harmonic oscillator angular frequency.
Notes
=====
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.abc import x, omega
>>> E_n(x, omega)
hbar*omega*(x + 1/2)
"""
return hbar * omega * (n + S.Half)
def coherent_state(n, alpha):
"""
Returns <n|alpha> for the coherent states of 1D harmonic oscillator.
See https://en.wikipedia.org/wiki/Coherent_states
Parameters
==========
n :
The "nodal" quantum number.
alpha :
The eigen value of annihilation operator.
"""
return exp(- Abs(alpha)**2/2)*(alpha**n)/sqrt(factorial(n))
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