Quantum Mechanics in 3D: Examples of Representations

1. Chapter 4 Quantum Mechanics in 3D

2. Examples of representations • Let us consider a set of functions: • The set is orthonormal: • Kets can be expanded:

3. Examples of representations • Closure relation

4. Examples of representations • For two kets • a scalar product is:

5. Examples of representations • Useful relationship:

6. Examples of representations • Let us consider a set of functions: • The set is orthonormal: • Kets can be expanded:

7. Examples of representations • Closure relation

8. Examples of representations • For two kets • a scalar product is:

9. Change of representations • Recall: • Choosing • we obtain:

10. 4.1 r and p operators • For • we obtain: • where • Similarly • “Vector” operator r:

11. 4.1 r and p operators • “Vector” operator p: • Then:

12. 4.1 r and p operators • Analogously: • Then:

13. 4.1 r and p operators • Calculating a commutator: • Similarly:

14. 4.1 r and p operators • Calculating a matrix element: • Thus: • Similarly: • Since |r > and |p > constitute complete bases operators r and p are observables

15. 4.3 Orbital angular momentum • In classical mechanics the angular momentum of a point mass relative to some axis is defined as: • In quantum mechanics the orbital angular momentum is defined by applying a quantization operation to the classical expression – replacing classical physical quantities with the corresponding observables (operators): • This definition does not require symmetrization with respect to non-commuting operators, e.g.:

16. 4.3 Orbital angular momentum • For a system of (spinless) particles the total orbital angular momentum is defined as:

17. 4.3 Commutation relations • Let us consider: • Similarly, one can obtain: • Thereby:

18. Levi-Civita symbol Tullio Levi-Civita (1873 – 1941)

19. Angular momentum • As we will see later, the so called spin can be treated as an intrinsic (non-orbital) angular momentum, satisfying similar commutation relations: • Thus both the orbital and intrinsic angular momenta are manifestation of the observable called angular momentumJ, defined via the commutation relations:

20. 4.3 Angular momentum • Let us introduce an operator: • This operator is Hermitian, and we will assume it is an observable

21. 4.3 Angular momentum • Let us consider: • Similarly, one can obtain: • Thereby:

22. 4.3 Angular momentum • What is the physical meaning of the commutation relations? • It is impossible to measure simultaneously the three components of the angular momentum, however, J2 and any component of J are compatible and could be measured simultaneously • Therefore, there is a possibility to find simultaneous eigenstates of J2 and any component of J (e.g., Jz)

23. 4.3 Angular momentum • We introduce (non-Hermitian) operators: • Let us consider: • Also

24. 4.3 Angular momentum • We introduce (non-Hermitian) operators: • Let us consider: • Synopsizing:

25. 4.3 Angular momentum • Let us also calculate: • Similarly: • Therefore:

26. 4.3 Eigenvalues • Let us consider an eigenvalue problem: • Recalling the expression • Moreover

27. 4.3 Eigenvalues • Let us consider an eigenvalue problem: • This also can be written as: • And this also can be written as:

28. 4.3 Eigenvalues • Let us consider an eigenvalue problem: • This also can be written as: • And this also can be written as:

29. 4.3 Eigenvalues • Since: • On the other hand, the square of the eigenvalue of the z-component of the angular momentum cannot exceed the eigenvalue of its magnitude squared: • Therefore, there should be top and bottom “rungs” for integers n and p

30. 4.3 Eigenvalues • Let us assume that for the top “rung” the eigenstate is: • Then: • And: • Recall:

31. 4.3 Eigenvalues • Let us assume that for the bottom “rung” the eigenstate is: • Then: • And: • Recall:

32. 4.3 Eigenvalues • Combining: • There are two solutions: • Thereby: • j must be integer or half-integer

33. 4.3 Eigenvalues • Using: • So: • We will therefore use indices j and m to label the eigenstates common to J2 and Jz

34. 4.3 Eigenvalues • Synopsizing: • We thus have found the eigenvalues of the angular momentum • What are the eigenstates?

35. 4.3 Eigenproblem for orbital momentum • We return to the orbital angular momentum of a spinless particle • Let us find relevant eigenstates in the r-representation • The Cartesian components of the orbital angular momentum operator:

36. 4.3 Eigenproblem for orbital momentum • It is convenient to work in spherical coordinates

37. 4.3 Eigenproblem for orbital momentum • Then

38. 4.3 Eigenproblem for orbital momentum • Then • And

39. 4.3 Eigenproblem for orbital momentum • Recall: • For the orbital angular momentum: • In the r-representation (and spherical coordinates):

40. 4.1 Eigenproblem for orbital momentum • In these equations r does not appear in the differential operators, so we will consider it as a parameter • Thus, the wavefunction can be written as: • We try separating the variables:

41. 4.1 Eigenproblem for orbital momentum • In these equations r does not appear in the differential operators, so we will consider it as a parameter • Thus, the wavefunction can be written as: • We try separating the variables:

42. 4.1 Eigenproblem for orbital momentum • In these equations r does not appear in the differential operators, so we will consider it as a parameter • Thus, the wavefunction can be written as: • We try separating the variables:

43. 4.1 Eigenproblem for orbital momentum • Then:

44. 4.1 Eigenproblem for orbital momentum • Then:

45. 4.1 Eigenproblem for orbital momentum • Then:

46. 4.1 Eigenproblem for orbital momentum • Then: • Since m is integer, l shoud be also an integer (not a half-integer)

47. 4.1 Adrien-Marie Legendre (1752 –1833) Eigenproblem for orbital momentum • We successfully separated variables but still need to solve this equation: • Solutions: • Here Pl are the Legendre polynomials:

48. 4.1 Adrien-Marie Legendre (1752 –1833) Eigenproblem for orbital momentum • Legendre polynomials:

49. 4.1 Eigenproblem for orbital momentum • The resulting solutions: • They have to be normalized • This yields: • Functions Y are called spherical harmonics

50. 4.1 Eigenproblem for orbital momentum • Spherical harmonics: