Spaces:
Sleeping
Sleeping
| 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)) | |