Angular Momentum in Quantum Mechanics

Transcription

1 Anguar Momentum in Quantum Mechanics Modern Physics Honor s Contract pring 007 Boone Drummond Mentor Dr. Cristian Bahrim 1

2 Contents Wave Characteristic of Eectron in Motion Anguar Momentum Overview Uncertainty Principe for Anguar Momentum Atomic Magnetic Momentum Intrinsic Anguar Momentum (pin) tern-gerach Experiment

3 Wave Characteristic of Eectrons in Motion Eary experimenta investigation of X-ray diffraction on crystas. Davisson and Kunsman (1914) shoot a beam of eectrons on a crysta. They thought that the eectron were scattering due to the crystas structure. de Brogie (191) hypothesis (diffraction) λ = Both parties stuck to their concusions and coud not agree on whether the eectrons were scattered or diffracted. Water Esasser (195) A student of Bohr pubished a paper on the eectron diffraction by a crysta using de Brogie hypothesis regarding the wave behavior of the eectrons in motion. chrodinger - pieced together a new theory of atomic structure which used de Brogie hypothesis quantum theory. h p r r p =h k The scientific community finay agreed on behaf of de Brogie. 3

4 Anguar momentum - Overview Orbita momentum - cassicay ur r ur = r p Projection aong an -axis = ur cosθ Orientation Quantum mechanics ur = ( + 1) h Quantum numbers = mh m =,...,0,... cosθ = = 0,..., n 1 r Z 4

5 pace quantiation certain vaues for θ =+ 3h =+ h = + 1h m =+ 1 = 0 m = 0 = 1h m = 1 = h m = = 3h m = 3 m = + 3 m = + cosθ = Exampe if then m ( + 1) =3 m = 0, ± 1, ±, ± 3 = 0, ± 1 h, ± h, ± 3h Therefore, there are 7 possibe orientations. 5

6 θ Uncertainty Principe is the ange between the vector anguar momentum and the -axis. This wi make a cone around the -axis. It is uncertain what is the exact orientation of the vector. The best we can know is a cone of high and side r. Therefore, the wave associated to an eectron of orbita anguar momentum r has a cyindrica symmetry. r x y are continuay changing because of the uncertainty principe. r p r Z Z = rpr ~h φ ~h 6

7 ur µ Atomic Magnetic Momentum U uuur B ext If there is an externa magnetic fied then the atomic magnetic momentum is couping with it. The energy that comes from this interaction is = µ B ext We take the -axis aong the externa magnetic fied. The atomic magnetic momentum aigned opposite to the fied has more energy then the one ined-up in the direction of the fied. Its magnitude can be obtained by the product of the current associated to the eectron in motion on a stationary state and the area of the orbit. µ = q ia = r π = T q = e q vr rπ Where q is the charge of the eectron and T is the period of revoution around the nuceus. T = π rπ v 7

8 ur Because of the uncertainty in the orientation of, the orientation of ur ur µ Consequences ur µ is aso unknown. The vectors and are aigned aong the -axis but are opposite to each other. or µ = q m vrm = ur e ur µ = m q m Discussion: Without an externa magnetic fied, There is one -state. This state has a (+1) degeneracy. Inside an externa magnetic fied, the degeneracy is removed and we see (+1) m states. m = 1 The subscript of is added to remind us that this magnetic momentum arises from the eectron s orbita anguar motion. =1 m = m = 1 0 8

9 r Intrinsic Anguar Momentum (pin) By using an anaogy with the orbita motion: we can define a new anguar vector: uur µ = ur e ur µ = m e ur m is named the intrinsic anguar momentum of the eectron and cassicay, represents the eectron s spinning motion about its own axis. If an externa magnetic fied exists then an interaction between µ r and B ext occurs. U = µ B ext 9

10 ur Quantum mechanicay is simiar with the orbita anguar momentum r. The difference is that it can have ony two projections. Anaogy: Think of the Earth s spinning around its own axis: it coud spin-up or down (these are the two directions). The ength of spin vector is given by the equation: 1 ± h ur = s( s + 1) h = s s = = m h is 1 3 h 4 where the spin quantum number is and the quantum number associated to the projection of the spin m s = ± 1 = + = s =1/ h h r ince there can ony be two orientations of the spin. = 3 4 h 10

11 Uncertainty principe Z φ ~h Z φ ~ h r 11

12 An atom in an externa magnetic fied without n B r ext 1 0 with m B r ext m s up down up down up down up down U Tota anguar momentum r J = r + r r r r µ + µ = µ atom r r r r = µ B µ B ext ext 1

13 tern-gerach Experiment In 191, two scientists by the name of W. Gerach and O. tern shoot a beam of siver atom through a non-uniform magnetic fied. They expected to see one signa due to the orbita motion of the eectron, ony. However, they observed two signas. This finding required the consideration of a new type of anguar motion for the eectron. o, they assumed the existence of the eectronic spin. 13

14 Exampe: How many signas coud have appeared in the tern- Gerach experiment? They used Ag (siver): Therefore: ur e ur µ = m the ground state has = 0 and m = ± e m µ = e = ( + 1) h = 0 m e eh µ = µ = ± m Z m 1 signas In genera for a fixed we shoud see ( + 1) 14

15 Bibiography Gribbin, John, In earch of chrodinger s Cat. New York: Bantam Books, 1984 Krane Kenneth, Modern Physics: econd Edition. John Wiey & ons Inc., Gerach_experiment.PNG/300px-tern- Gerach_experiment.PNG 15