Sunday, February 13, 2011

Mastering Physics: The Bohr Atom

Learning Goal: To understand the Bohr model of the hydrogen atom.
In 1913 Niels Bohr formulated a method of calculating the different energy levels of the hydrogen atom. He did this by combining both classical and quantum ideas. In this problem, we go through the steps needed to understand the Bohr model of the atom.

Part A
Consider an electron with charge -e and mass m orbiting in a circle around a hydrogen nucleus (a single proton) with charge +e. In the classical model, the electron orbits around the nucleus, being held in orbit by the electromagnetic interaction between itself and the protons in the nucleus, much like planets orbit around the sun, being held in orbit by their gravitational interaction. When the electron is in a circular orbit, it must meet the condition for circular motion: The magnitude of the net force toward the center, F_c, is equal to mv^2/r. Given these two pieces of information, deduce the velocity v of the electron as it orbits around the nucleus.

Express your answer in terms of e, m, r, and epsilon_0, the permittivity of free space.

  v  = \frac{e}{\sqrt{mr4{\pi}{\epsilon}_{0}}}

Part B
The key insight that Bohr introduced to his model of the atom was that the angular momentum of the electron orbiting the nucleus was quantized. He introduced the postulate that the angular momentum could only come in quantities of nh/(2 \pi), where h is Planck's constant and n is a nonnegative integer (0, 1, 2, 3, \dots). Given this postulate, what are the allowable values for the velocity v of the electron in the Bohr atom? Recall that, in circular motion, angular momentum is given by the formula L=mvr.
Express your answer in terms of n, Planck's constant h, m, and r.

  v  = \frac{nh}{2{\pi}mr}

Part C
In Parts A and B you found two different expressions to describe the allowed electron velocities v. Equate these two values (eliminating v) and solve for the allowable radii r in the Bohr model.
Express r in terms of n, h, m, e, and epsilon_0.

  r  = \frac{n^{2}h^{2}{\epsilon}_{0}}{{\pi}me^{2}}
 As the electron orbits the nucleus with a speed v at a radius r, it has both kinetic energy and electric potential energy. The total energy E of the electron can be expressed as
E=\frac {1}{2} m v^2 - \frac{e^2}{4\pi \epsilon_0 r}.

Part D
In Parts B and C you saw that, according to Bohr's postulate, the electron radius r and the electron velocity v only have certain allowable values. Plug the values obtained for these two quantities into the energy statement given above to arrive at a new statement for the allowed energy levels in the Bohr atom.
Express your answer in terms of e, m, n, h, and epsilon_0.

  E  = -\frac{e^{4}m}{8n^{2}h^{2}{\epsilon}_{0}{^{2}}}

The standard formula for energy in the Bohr model is
E_n = - \frac{e^4m}{8n^2h^2\epsilon_0^2}.
Despite the serious flaws of the Bohr model, such as casually mixing quantum and classical ideas with little if any justification, this formula turns out to be equivalent to the energy formula for hydrogen obtained from quantum mechanics. To adequately deal with atoms other than hydrogen, however, requires the full quantum theory.

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