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