Lecture 13 Space quantization and spin

1. Lecture 13Space quantization and spin (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies.

2. z-component angular momentum • The z-component of the angular momentum operator depends only on φ • The “particle on a sphere” wave function is the eigenfunction of the lz operator. Angular momentum

3. Total angular momentum • The total angular momentum operator is

4. Total angular momentum • The “particle on a sphere” wave function is also the eigenfunction of the l2totaloperator. Angular momentum squared

5. z-component and total angular momenta • The “particle on a sphere” wave function, therefore, has well-defined total energy and total angular momentum: • … and well-defined z-component energy and z-component angular momentum:

6. x- and y-component angular momenta • This is because lzand ltotal commute. • lxand ly operators depend on both θ and φ. They do not commute with lz or ltotal or H.

7. Uncertainty principle

8. Space quantization • If all of x, y, and zcomponents were known, we knew the angular momentum vector (length and direction) exactly and hence the circular trajectory perpendicular to it. • The uncertainty in x and y components indicates the precise trajectory cannot be known.

9. Space quantization • We can only know the total ( )and z component ( ). • x and y components remain undetermined, so we do not know the precise trajectory.

10. Space quantization • Even when ml = l, z-component does not exhaust the total angular momentum because. • If it were not for “+1”, ml = lwould leave nothing for x and y components and precisely determine all x, y,and z components simultaneously, violating uncertainty principle!

11. Angular momentum as a magnet • Rotational motion of a charged particle (such as an electron) gives rise to a magnetic field. • Angular momentum is proportional to the magnetic moment. • Applying an external magnetic field (along z axis) and measuring the interaction, one can determine the (z-component) angular momentum of an electron.

12. Stern-Gerlach experiment • The trajectories of electrons in an inhomogeneous magnetic field are bent. • The trajectories are “quantized” – the proof of the quantization of angular momentum orientation, namely, space quantization.

13. Summary • The total angular momentum and only one of the three Cartesian components (z-component) can be determined exactly simultaneously. One component cannot exhaust the total momentum (because of “+1” in ). • The angular momentum is quantized in both its length and orientation – it cannot point at any arbitrary direction (space quantization).

14. Stern-Gerlach experiment • Just two trajectories observed for electrons. • This suggests l = ½ and m =½ and –½. • This cannot exactly be a particle on a sphere (where l and m must be full integers)?

15. Spin • It is the “spin” angular momentum of an electron. • It has been discovered that the particle has intrinsic magnetic momentum. Its precise derivation is beyond quantum chemistrty. • We can imagine the particle spinning and its associated angular momentum acts like a magnetic moment.

16. Spin • An electron has the spin quantum numbers = ½ (corresponding to l of particle on a sphere). The total spin angular momentum is . • The spin magnetic quantum number ms (corresponding to ml) can take s,…,–s (unit interval). For an electron ms= ½ (spin up or α spin) and –½ (spin down or β spin). • A proton or neutron also has s = ½. • A photon has s = 1.

17. Indistinguishable particles • Because of the uncertainty in the position and momentum of a particle, in a microscopic scale, two particles of the same kind (such as two electrons) nearby are indistinguishable.

18. Indistinguishable particles • The probability of finding particle 1 at position r1 and particle 2 at r2 is the same as that of finding particle 2 at r1 and particle 1 at r2 (otherwise we can distinguish the two). There are at least two immediate possibilities

19. Fermions • Possibility 1:This is the case with particles having half integer spin quantum numbers (such as electrons). They are called fermions. • Two fermions cannot occupy the same position in space (Pauli exclusion principle). They form matter.

20. Bosons • Possibility 2:This is the case with particles having full integer spin quantum numbers (such as photons). They are called bosons. • Two bosons can occupy the same position in space (e.g., photons can be superimposed to become more intense). They tend to mediate fundamental interactions.

21. Caveat • The derivation of the concepts of spin, fermions, and bosons is beyondquantum chemistry. • However, spin and the Pauli exclusion principle for electrons (as fermions) are a critical element of chemistry (whenever we have more than one electrons). • We treat these as external postulates to quantum chemistry.

22. Homework Challenge #3 • Find the origin of spins. Explain why the spin quantum number of an electron, proton, and neutron is ½, whereas that of a photon is 1. • Explain why particles with half integer spin quantum numbers are fermions and have wave functions that are anti-symmetric with respect to particle interchange. Explain why particles with full integer spin quantum numbers are bosons and their wave functions are symmetric with respect to interchange.