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from sympy.core import S, pi, Rational |
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from sympy.functions import hermite, sqrt, exp, factorial, Abs |
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from sympy.physics.quantum.constants import hbar |
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def psi_n(n, x, m, omega): |
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""" |
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Returns the wavefunction psi_{n} for the One-dimensional harmonic oscillator. |
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Parameters |
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========== |
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n : |
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the "nodal" quantum number. Corresponds to the number of nodes in the |
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wavefunction. ``n >= 0`` |
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x : |
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x coordinate. |
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m : |
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Mass of the particle. |
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omega : |
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Angular frequency of the oscillator. |
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Examples |
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======== |
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>>> from sympy.physics.qho_1d import psi_n |
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>>> from sympy.abc import m, x, omega |
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>>> psi_n(0, x, m, omega) |
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(m*omega)**(1/4)*exp(-m*omega*x**2/(2*hbar))/(hbar**(1/4)*pi**(1/4)) |
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""" |
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n, x, m, omega = map(S, [n, x, m, omega]) |
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nu = m * omega / hbar |
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C = (nu/pi)**Rational(1, 4) * sqrt(1/(2**n*factorial(n))) |
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return C * exp(-nu* x**2 /2) * hermite(n, sqrt(nu)*x) |
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def E_n(n, omega): |
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""" |
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Returns the Energy of the One-dimensional harmonic oscillator. |
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Parameters |
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========== |
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n : |
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The "nodal" quantum number. |
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omega : |
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The harmonic oscillator angular frequency. |
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Notes |
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===== |
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The unit of the returned value matches the unit of hw, since the energy is |
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calculated as: |
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E_n = hbar * omega*(n + 1/2) |
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Examples |
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======== |
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>>> from sympy.physics.qho_1d import E_n |
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>>> from sympy.abc import x, omega |
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>>> E_n(x, omega) |
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hbar*omega*(x + 1/2) |
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""" |
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return hbar * omega * (n + S.Half) |
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def coherent_state(n, alpha): |
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""" |
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Returns <n|alpha> for the coherent states of 1D harmonic oscillator. |
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See https://en.wikipedia.org/wiki/Coherent_states |
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Parameters |
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========== |
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n : |
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The "nodal" quantum number. |
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alpha : |
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The eigen value of annihilation operator. |
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""" |
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return exp(- Abs(alpha)**2/2)*(alpha**n)/sqrt(factorial(n)) |
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