Atomic Physics 2

1. Atomic Physics 2

2. Topics • Recap • Quantization of Energy • Quantum States of Hydrogen • Spin and the Pauli Exclusion Principle • Summary

3. Recap The wave functions for the hydrogen model are of the form where the Y(q,f) are the spherical harmonics and R(r) is the radial function. The quantum numbers l and m describe the angular shape of the wave functions

4. Recap m = 0 1 2 3 l = 0 The square of the first few spherical harmonics l = 1 l = 2 l = 3 http://mathworld.wolfram.com/SphericalHarmonic.html

5. Quantization of Energy The quantization of angular momentum occurs in any system that is spherically symmetric and has the same form: The energy is also quantized, but how depends on the precise form of the potential V(r)

6. Quantization of Energy For the hydrogen atom, the solution of the Schrödinger equation yields the same answer as that obtained using the Bohr model Z = 1 where

7. Quantization of Energy The exact form of the radial function R(r) can be obtained for the hydrogen atom and is found to depend on the orbital quantum number l. R(r) also depends on another integer n, called the principal quantum number, which is the same integer that appears in the result for the quantized energy.

9. Radial Functions of Hydrogen n = 3 radial functions http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/HydrogenAtom.htm

10. Quantum States of Hydrogen We have seen that three quantum numbers n, l, m are needed to specify a quantum state of the hydrogen model Long before the quantum theory, these numbers were used to classify spectral lines (but with no understanding of their meaning)

11. Quantum States of Hydrogen Even though we now understand the meaning of the quantum numbers, we still use the old spectroscopic code to label the states of hydrogen. The code uses n and a letter: S for l = 0 P for l = 1 D for l = 2 F for l = 3 G for l = 4

12. Quantum States of Hydrogen Energy, eV -3.4 eV Energy level diagram of hydrogen and transitions Transition rule -13.6 eV

13. Electron Spin When viewed with high precision the energy levels of hydrogen contain fine structure that is not explained by the theory we have outlined so far. In 1925, in order to explain the fine structure, Wolfgang Pauli suggested that an electron has a 4th quantum number that takes on just 2 values.

14. Electron Spin Later that year, the Leiden graduate students S. Goudsmit and G. Uhlenbeck suggested that Pauli’s quantum number could be the z component of an intrinsic angular momentum of the electron, which they called spin From the viewpoint of the electron, the “motion” of the nuclear charge generates a magnetic field with which the electron spin interacts, thereby creating the observed fine structure

15. The Pauli Exclusion Principle As far as we know, all electrons are identical. Therefore, electrons are indistinguishable. So too are photons of the same wavelength For hydrogen the total wave function is where X is the spin wave function and ms is the spin quantum number (with values ±1/2)

16. The Pauli Exclusion Principle The Pauli exclusion principle states that no more than one electron can occupy a given quantum state It is no exaggeration to state that atoms, and therefore structure, would not exist if this principle were false. Five billion years from now the Sun will become a white dwarf and the Pauli exclusion principle will be crucial to its stability!

17. Summary • In atoms the magnitude and z component of angular momentum are quantized • The energy is also quantized • The electron has an intrinsic attribute called spin, which is responsible for the observed fine structure in spectra • Exclusion principle: Only one electron can be in a given quantum state