3. The transition dipole moment defined in class can be used to determine select

Question

3. The transition dipole moment defined in class can be used to determine selections rules of electronic transitions in atoms. a. Calculate the transition dipole moment, 42) where --er cos , for a transition from the 1s to the 2s level in the hydrogen atom. Show that this transition is forbidden, i.e., that the integral above is zero. b. Calculate the transition dipole moment defined above for a transition from the 1s level to the 2p-level in the hydrogen atom. Show that this transition is allowed, i.e., that the integral above is not zero. Use 32T a for the 2p wave function. c. Comment on why we can observe transitions between energy levels n - 1 and n- 2 for the hydrogen atom, and qualitatively discuss the effect of a magnetic field oriented along z on the spectrum of this transition.

Explanation / Answer

Answer:

Part(C)

According to the theory quantum mechanics, an electron can not have any value of energy,which is bound to an atom, rather it can only occupy certain states which correspond to certain energy levels. The formula for the energy levels of a Hydrogen atom are given by the equation: E = -Eo/n2, where Eo = 13.6 eV (1 eV = 1.602×10-19 Joules) and n = 1,2,3… and so on.
The energy is expressed as a negative number because it takes that much energy to ionize the electron from the nucleus. Because an electron bound to an atom can only have certain energies the electron can only absorb photons of certain energies which matches exactly to the energy difference, or “quantum leap”, between two energy states.
When an electron absorbs a photon it gains the energy of the photon. Because an electron bound to an atom can only have certain energies the electron can only absorb photons of certain energies.
For example, an electron in the ground state has an energy = -13.6 eV. The second energy level = -3.4 eV. Thus it would take E2 E1= -3.4 eV -13.6 eV = 10.2 eV to excite the electron from the ground state to the first excited state.

Effect of magnetic field is also called as the Zeeman effect. Now,let us predict how Zeeman effect for the 2p to 1s transition in hydrogen would look like, and then compare this prediction with a more complete theory. To understand the Zeeman effect, which uses a magnetic field to remove the degeneracy of different angular momentum states, we need to examine how an electron in a hydrogen atom interacts with an external magnetic field, B. Since magnetism results from the circular motion of charged particles, we should look for a relationship between the angular momentum, L and the magnetic dipole moment, m

The relationship between the magnetic dipole moment m and the angular momentum L of a particle with mass m and charge q is given by
m=(q/2m)L

For electron the equation becomes,(-e/2me)L
where, the specific charge and mass of the electron have been substituted for q and m. The magnetic moment for the electron is a vector pointing in the direction opposite to L, both of which classically are perpendicular to the plane of the rotational motion.

NOTE: Due to magnetic field along z axis spin occurs.For a spin I = 1/2 nucleus in a magnetic field of strength Bo, the nucleus' magnetic moment will precess about the z-axis defined by the direction of the magnetic field. The nuclei may be oriented either parallel or antiparallel to the direction of the magnetic field.Consequently, some spins orient about the positive z-axis and vice versa.

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