95 lines
2.2 KiB
Python
95 lines
2.2 KiB
Python
from sympy.core.numbers import (I, pi)
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from sympy.core.singleton import S
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from sympy.functions.elementary.exponential import exp
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.physics.quantum.constants import hbar
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def wavefunction(n, x):
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"""
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Returns the wavefunction for particle on ring.
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Parameters
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==========
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n : The quantum number.
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Here ``n`` can be positive as well as negative
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which can be used to describe the direction of motion of particle.
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x :
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The angle.
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Examples
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========
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>>> from sympy.physics.pring import wavefunction
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>>> from sympy import Symbol, integrate, pi
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>>> x=Symbol("x")
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>>> wavefunction(1, x)
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sqrt(2)*exp(I*x)/(2*sqrt(pi))
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>>> wavefunction(2, x)
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sqrt(2)*exp(2*I*x)/(2*sqrt(pi))
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>>> wavefunction(3, x)
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sqrt(2)*exp(3*I*x)/(2*sqrt(pi))
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The normalization of the wavefunction is:
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>>> integrate(wavefunction(2, x)*wavefunction(-2, x), (x, 0, 2*pi))
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1
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>>> integrate(wavefunction(4, x)*wavefunction(-4, x), (x, 0, 2*pi))
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1
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References
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==========
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.. [1] Atkins, Peter W.; Friedman, Ronald (2005). Molecular Quantum
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Mechanics (4th ed.). Pages 71-73.
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"""
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# sympify arguments
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n, x = S(n), S(x)
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return exp(n * I * x) / sqrt(2 * pi)
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def energy(n, m, r):
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"""
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Returns the energy of the state corresponding to quantum number ``n``.
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E=(n**2 * (hcross)**2) / (2 * m * r**2)
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Parameters
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==========
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n :
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The quantum number.
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m :
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Mass of the particle.
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r :
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Radius of circle.
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Examples
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========
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>>> from sympy.physics.pring import energy
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>>> from sympy import Symbol
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>>> m=Symbol("m")
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>>> r=Symbol("r")
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>>> energy(1, m, r)
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hbar**2/(2*m*r**2)
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>>> energy(2, m, r)
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2*hbar**2/(m*r**2)
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>>> energy(-2, 2.0, 3.0)
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0.111111111111111*hbar**2
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References
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==========
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.. [1] Atkins, Peter W.; Friedman, Ronald (2005). Molecular Quantum
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Mechanics (4th ed.). Pages 71-73.
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"""
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n, m, r = S(n), S(m), S(r)
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if n.is_integer:
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return (n**2 * hbar**2) / (2 * m * r**2)
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else:
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raise ValueError("'n' must be integer")
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