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chapter 5

chapter 5. Angular Momentum. outline. simultaneous specification of several properties vectors angular momentum of a one-particle system the ladder-operator method for angular momentum summary. simultaneous specification of several properties. Postulation(from3.3):

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chapter 5

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  1. chapter 5 Angular Momentum

  2. outline • simultaneous specification of several properties • vectors • angular momentum of a one-particle system • the ladder-operator method for angular momentum • summary

  3. simultaneous specification of several properties • Postulation(from3.3): • If the state function is an eigenfunction of the operator A with eigenvalue s, then a measurement of the physical property A is certain to yield the result s.

  4. Simultaneous Specification of Several Properties • When will it be possible for to be simultaneously an eigenfunction of two different operator? • In chapter 7 • A necessary condition for the existence of complete set of simultaneous eigenfunctions of two operators is that the operators commute with other. • If A and B are two commuting operator that correspond to physical quantities, then there exists a complete set of functions that are eigenfunctions of both A and B. • Thus,if ,then f can be an eigenfunction of both A and B .

  5. Commutator identities

  6. Example • Use the commutator identities to find for a one-particle,three-dimensional system. • (a)

  7. 0

  8. Simultaneous Specification of Several Properties • We cannot expect the state function to be simultaneously an eigenfunction of and of . • Hence we cannot simultaneously assign definite values to and . • In agreement with the uncertainty principle.

  9. and do not commute →we cannot expect to assign definite values to the energy and the x coordinate at the same time. For a state function that is not an eigenfunction of ,we get various possible outcomes when we measure A in identical systems. We want some measure of the spread or dispersion in the ser of observed values Ai. Variance： Simultaneous Specification of Several Properties

  10. (ch7) ↓ 1 ↓ 1 Simultaneous Specification of Several Properties Heisenberg uncertainty principle For the product of the standard deviations of two properties of a quantum-mechanical system whose state function is ,one can show that

  11. Simultaneous Specification of Several Properties

  12. Simultaneous Specification of Several Properties • shows that Δ t is to be interpreted as the lifetime of the state whose energy is uncertain by ΔE. • It is often stated that Δ t is the duration of the energy measurement. • Aharonov and Bohm have shown that “energy can be measured reproducibly in an arbitarily short time.

  13. Simultaneous Specification of Several Properties

  14. Simultaneous Specification of Several Properties

  15. Problem

  16. Example

  17. vectors • dot product and cross product • del , gradient , curl , and divergence • n-dimensional vector Space • coordinate systems

  18. C A A A B B B Vector addition

  19. The product of a vector and a scalar,cA,is defined as a vector of length ∣c∣times the length of A with the same direction as A if c is positive,or the opposite direction to A if c is negative.

  20. z Az A y k Ay i Ax j x Unit vectors i,j,k,and components of A • To obtain an algebraic way of representing vectors, we set up Cartesian coordinates in space.

  21. A θ B Dot Product (distributive law)

  22. 1 Dot Product • Dot product of a vector with itself

  23. k j i Cross Product

  24. k j A×B i B θ A Cross Product

  25. 0 k -j -k 0 j -i 0 i Cross Product

  26. coordinate systems • Cartesian coordinate • Cylindrical coordinate • Spherical coordinate

  27. z Az A y k Ay i Ax j x Cartesian coordinate

  28. Cylindrical coordinate

  29. Spherical coordinate

  30. del , gradient

  31. n-dimensional vector Space

  32. n-dimensional vector Space

  33. z θ1 y θ1 x θ2 Direction angles • A two-dimensional vector can be specified by its length and one direction angles. • A three-dimensional vector can be specified by its length and two direction angles. • An n-dimensional vector can be specified by its length and n-1 direction angles.

  34. angular momentum of a one-particle system • Classical mechanics of one-particle angular momentum

  35. angular momentum of a one-particle system • One-particle angular momentum operators • orbital angular momentum • spin angular momentum (ch 10)

  36. orbital angular momentum • Commutation relations determine which physical observables can be simultaneously assigned definite values, so we exam these relations for angular momentum.

  37. We can specify an exact value for and any one component. • We can’t specify more than one component simultaneously. • Note that in specifying we are not specifying the vector ,only its magnitude.

  38. Transform the angular-momentum operator to spherical coordinates

  39. One-particle orbital-angular-momentum eigenfunctions and eigenvalues

  40. the ladder-operator method for angular momentum

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