Question

3. Prior to quantum mechanics the Bohr Model viewed the atom as having electrons traveling in circular orbits (shells) about the positively charged nucleus. This model gives a reasonable estimate of the dipole moment of the hydrogen atom if one assumes the radius of the electron orbit about the nucleus is 5.3 x 10-11 m. Using the Bohr Model

(a) what is the dipole moment of the hydrogen atom? Now place the hydrogen atom in a magnetic field, B, with the plane of the electron orbit perpendicular to B. If the centripetal force on the electron is dominated by the electric field as opposed to the magnetic field we may assume that the orbit radius r does not change when the atom is placed in the magnetic field.

(b) If the electron circulates counterclockwise will an observer looking in the direction of B see the angular frequency of the electron decrease or increase – justify your answer.

(c) What if the electron is circulating clockwise?

(d) What is the electrons rotational frequency when B = 0?

(e) Show that the change in rotational frequency, Df, between when B is present and not present is approximately ± Be / 4p m.

Here e and m are the charge and mass of the electron respectively and we are ignoring terms of order B2 and higher. These frequency shifts are caused by the splitting of the energy levels of optical lines of excited atoms in a magnetic field, and were first observed by Pieter Zeeman (a Dutch Physicist) in 1896. Hendrik Lorentz (another Dutch physicist) shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of this phenomenon, now referred to as the Zeeman effect. The Zeeman effect underpins applications such a Magnetic Resonance Imaging (MRI) in which case such splitting is in nuclear energy levels as opposed to the atomic levels of concern here.

Answer 1