Spectra of Atoms When an atom is excited, it emits light. But not in the continuous spectrum as blackbody radiation! The light is emitted at discrete wavelengths. The set of wavelengths emitted by atoms of a given element is absolutely the same for every atom of that element, no matter how hot or cold it is. Mystery #4. What is the dynamics that causes line spectra?

Emission spectrum from hydrogen n 0 =1 n 0 =2 n 0 =3

How to explain these numerical series? Before we tackle that mystery, consider another: Mystery #5: Why don’t the electrons in an atom spiral into the nucleus? In classical electrodynamics, an accelerated charged particle radiates electromagnetic waves: When an energetic (free) electron scatters from a nucleus, it emits Bremmstrahlung radiation. When a charged particle is bent along a curved trajectory in a magnetic field, it emits synchrotron radiation.

Bohr’s postulate: angular momentum must be quantized! Calculate the kinetic energy associated with the orbit of an electron in a hydrogen atom, using classical mechanics: First write the orbit equation: Now extract the angular momentum: Express orbit radius in terms of L: Now quantize L: The orbit radius can only have discrete values:

First the potential energy: Now the kinetic energy: Finally the total energy: Now insert the quantized radius: And recover the series spectra! Now calculate the quantized energy levels

This is a remarkable result! The quantum of angular momentum is the same as the proportionality constant in the relation of energy and (angular) frequency: Angular momentum now plays a very special role in quantum physics: it can only take on values that are integer multiples of this basic unit value.

Let’s work a few problems 4.6 The energy loss per unit time due to an acceleration a acting on an electron is given by the Larmor formula: a) What would be the power loss for an electron in the first Bohr orbit in a hydrogen atom if it were able to lose energy by this classical process? First calculate the (centripetal) acceleration of the electron:

Use orbit equation to express a in terms of mass, potential energy, and radius:

b)Estimate the time it would take the electron to spiral into the nucleus if it able were to lose energy in this way. We must realize that, as it spirals in, the radius gets smaller and smaller. The acceleration a depends upon the radius, and therefore so does P: We must also realize that the radius depends upon the energy E: So the change in radius is

We need to solve this as a differential equation: The time to collapse to zero radius is then

4.18 a) Calculate the first three energy levels of Li ++ Li ++ has Z=3 and a single electron (it is a hydrogenic atom). b) What is the ionization potential of Li ++ ? The ionization potential is the energy difference between the ground state level and the large-n limit:

c) What is the first resonance potential for Li ++ ? The first resonance potential is the energy needed to excite the electron from the ground state to the first excited state: This is the first (lowest energy) line in the Lyman series of emission spectral lines.