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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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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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