Eigenvalues of the Angular Momentum Operators

Transcription

1 Eigenvalues of the Angular Moentu Operators Toda, we are talking about the eigenvalues of the angular oentu operators. J is used to denote angular oentu in general, L is used specificall to denote orbital angular oentu, and S is used to denote spin. J= L+S [Note: follows Section 9., page 358] J is generalied; we still have the couter relations:, i J, i J J L S total oent x, x i J Toda, we want to find the eigenvalues of J and J. The generalied relation is applicable to the axis. We will show that: J where, 1,...,0, 1, n ( 1) where and 0,1,,... J n Copare this to the orbital angular oentu operators: L where, 1,...,0, 1, ( 1) where 0,1,,... L l l l Ladder Operators Fro the coutator relations and fro the creation/annihilation operators fro the Haronic oscillator, we have the Ladder Operators for angular oentu. J J ij J J ij x x coutes with the otehr coponents of angular oentu J,,, 0 Soe other properties of these operators:, J We want to prove this directl., J

2 Here is our proof: We show this explicitl:,, ij x J J ij J ij J x J J ij J J J ij J x x x,, i x i J i J x J J ij We can change the order to have a new relation: J J J J J x Show that J J J J J We want to prove this relation directl. J J J ij J ij x x J J J ij ij J ij ij x x x x J ij J ij J J x x x J J i[ J, J ] x x J J J J J J J Prove the Coutator Relations We can use these relations put directl to find the coutator relations. J J J J J

3 , J J, J J J J J J ij J ij J ij J ij x x x x J J J ij ij J ij ij J J J ij ij J ij ij x x x x x x x x i J J J J x x i J, J x J, J i J x J, J i J x Also we can show: J J J J J J J J J J J J J J J J J J J J J Eigenvalues of Ĵ and J To solve the proble of Ĵ and J to solve eigenvalue of these two phsical paraeters first we see that the coute to each other., 0 Here is the eigenvalue equation: J State function of J eigenvalue

4 Use the coutator relations to change the order of these operators., J J J J J J J J J J 1 J 1 J J J So this iplies that: J 1 J J 1 1 Siilarl:, J J J J J J J J J J 1 J 1 J J J So this iplies that: J 1 J J 1 1 Now we need to find the coon eigenfunctions the eigenfunction of J and the eigenfunction k of Ĵ. Coon functions found fro: J k where k is an real integer k We calculate the expectation value: J J dx k also: 0 0 k if k 0 then k k ax ax ax we have this confineent has a axiu and also a iniu J 0 and J 0 J J J J J k k in and J k in in

5 Integral nuber of states To find the eigenvalues of Ĵ and J we use the coutator relations J J J J J : J J J J J 0 1 ax ax ax ax ax ax ax ax J k ax ax 1 k ax ax ax k ax ax 1 Look at this using the sae ethod to do iniu. J J J 0 1 J in in in in in in in in k in in 1 k in in in in k in in 1 Let: ax Fro the eigenfunction ou have J. Fro the ladder operator, we, 1,..., 1,. Here are the values: have the range of values ax ax -1 1 =0-1 in +1 in Here are the eigenvalues of Ĵ and J. We got this result fro the coutator relations J J J J J.

6 J n n 1 0,1,,... J, 1,...,0,..., 1, n=4 = n=3 =3/ n= =1 n=1 =1/ n=0 =0 Siilar to figure 9.6 on page 36. = =1 =0 =-1 =- =3/ =1/ =-1/ =-3/ =1 =0 =-1 =1/ =-1/ =0

7 Rotational Energ Levels of Molecules The rotational level of a olecule is the lowest energ level is rotation. Fro classical echanics: L I H and I a M a M We are not considering vibration, so a is constant. The eigenvalues are: L l l Ma E l l l l l 1 1 I 4 E E 1 4 l l l l l l Ma 4Ma l l l l Ma Using the sae ethod, sae eigenvalue equation talk about Hdrogen the siplest olecule M-M and the ΔE=E l -E l-1. Here 13.6 ev is a e Bohr called the Rdberg constant. It is the energ of ioniing a hdrogen ato. If we consider this Rdberg constant to be Ma is 13.6 ev M e where e M is usuall on the order of So the Rdberg constant is about 10eV for Hdrogen (about 1K energ and λ. Reeber for the qualifing exa if ou know this ev. Hoework: 9.5 and 9.6 not difficult siple like general phsics.