89 lines
2.0 KiB
Python
89 lines
2.0 KiB
Python
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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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# sympify arguments
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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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# normalization coefficient
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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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