109 lines
2.0 KiB
Python
109 lines
2.0 KiB
Python
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#!/usr/bin/env nix-script
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#!>python3
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#! python3 | sympy
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from sympy.physics.units import *
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from sympy import symbols, solve, Eq, pi
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##
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## Bohr model for the hydrogen atom
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##
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# Constants
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ε_0, e, m_e, h, c, π = symbols('ε_0 e m_e h c π')
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values = {
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e : 1.602176565e-19 * C,
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m_e: 9.10938291e-31 * kg,
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h : planck,
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ε_0: electric_constant,
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c : speed_of_light,
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π : pi }
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# Variables
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v, r, λ, ν, n = symbols('v r λ v n')
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##
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## Electron velocity
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##
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# Coulomb force is a centripetal force
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F_c = m_e * v**2/r
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F = 1/(4*π*ε_0) * e**2 / r**2
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# Equate and solve for the velocity (taking the positive solution)
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v = solve(Eq(F_c, F), v)[1]
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##
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## Energy levels
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##
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# Total energy of the electron
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U = -e**2 / (4*π * ε_0 * r)
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K = m_e * v**2 / 2
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E = U + K
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# De Broglie wavelength of the electron
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λ = h / (m_e*v)
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# Hypothesize the orbits are quantized
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C_o = 2*π * r
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C_q = n * λ
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# Solve for the radius
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r_n = solve(Eq(C_o, C_q), r)[0]
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# Bohr radius is the radius of the first orbit
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r_1 = r_n.subs(n, 1)
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# Substitute r_n in the energy equation
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E_n = E.subs(r, r_n)
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# The ground state is lowest energy level
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E_1 = E_n.subs(n, 1)
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##
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## Spectum of emission
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##
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# Energy of the emitted photon from Planck–Einstein relation
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λ = symbols('λ')
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ν = c/λ
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E_γ = h*ν
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# Energy radiated by the electron
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# for n > m
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E_n # initial state
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E_m = E_n.subs(n, m) # final state
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E = E_n - E_m
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# Equate and solve for λ
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λ = solve(Eq(E_γ, E), λ)[0]
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# Find the inverse wavelength
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λ_1 = 1/λ
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# Substitute the Rydberg constant
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R_h = e**4 * m_e / (8 * c * ε_0**2 * h**3)
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λ_1 = λ_1.subs(R_h, symbols('R_h'))
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value = lambda x: x.subs(values).evalf(10)
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results = '''
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Bohr radius: {r_1} ≈ {vr_1}
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Ground state / Rydberg energy: {E_1} ≈ {vE_1}
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Rydberg constant: {R_h} ≈ {vR_h}
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Energy of the nth level: E_n = {E_n}
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Rydberg equation: 1/λ = {λ_1}
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''' .format(vr_1=value(r_1),
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vE_1=value(E_1),
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vR_h=value(R_h),
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**locals()
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).replace('**','^').replace('*', ' ')
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print(results)
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