{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter4_2 - h h = − = = − − − − = − − = −...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
PHY6404 R. D. Field Department of Physics Chapter4_2.doc University of Florida The Angular Momentum Operator (1) The Momentum Operator in 3 Dimensions: In “position space” with Cartesian coordinates we have z i p y i p x i p op z op y op x = = = h h h ) ( ) ( ) ( and 2 2 2 2 2 2 ) ( ) ( ) ( ) ( op op z op y op x op p p p p = + + = h where 2 2 2 2 2 2 2 z y x op + + = . Angular Momentum: Angular momentum is the vector operator given by op z op y op x op op p p p z y x z y x p r L ) ( ) ( ) ( ˆ ˆ ˆ = × = r r r Hence, op y op z op x p z p y L ) ( ) ( ) ( = op z op x op y p x p z L ) ( ) ( ) ( = op x op y op z p y p x L ) ( ) ( ) ( = and in “position space” with Cartesian coordinates we have = y z z y i L op x h ) ( = z x x z i L op y h ) ( = x y y x i L op z h ) ( Commutation Relations: The commutator of, for example, L x and L y is ( ) op z op x op y op y op z op x op z op y op z op z op y op y op x op x op y op z op z op z op z op z op x op x op z op z op y op x op y op z op z op x op z op z op x op y op z op y op x L i p y p x i p p z x p z p y p p x z p x p z p p z z p z p z p p x y p x p y p p z y p z p y p x p z p z p z p x p y p z p y p x p z p z p y L L ) ( ) ( ) ( ) ]( ) ( , [ ) ]( , ) [( ) ]( ) ( , [ ] ) ( , ) [( ) ]( ) ( , [ ] ) ( , ) [( ) ]( ) ( , [ ] ) ( , ) [( ) ]( ) (
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ( , [ ] ) ( , ) [( ] ) ( , ) ( [ ] ) ( , ) ( [ ] ) ( , ) ( [ ] ) ( , ) ( [ ] ) ( ) ( , ) ( ) ( [ ] ) ( , ) [( h h = − = + = + + − − − − + = + − − = − − = We see that the commutator of any two of the angular momentum operators gives the third angular momentum operator as follows: op z op y op x L i L L ) ( ] ) ( , ) [( h = op x op z op y L i L L ) ( ] ) ( , ) [( h = op y op x op z L i L L ) ( ] ) ( , ) [( h = This can be summarized by the following: op k ijk op j op i L i L L ) ( ] ) ( , ) [( ε h = Note: ε iik = ε ijj = ε iji = 0, ε 123 = ε 231 = ε 312 = 1, ε 213 = ε 132 = ε 321 = -1....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern