Question

Consider a particle with angular momentum ( j=1 ). (a) Write down the three basis states which are eigenvectors of ( J_{z} ) in both Dirac notation and as vectors. [1] (b) Write the matrices for the operators ( J^{2}, J_{+}, J_{-}, J_{x}, J_{y} ), and ( J_{z} ). Which of these operators are Hermitian? ( [15] ) (c) Which operators in part ( (b) ) form a complete set of commuting observables? [1] (d) Consider the Hamiltonian ( H=alpha J_{z} ) ( ( alpha ) is a real number). What are the energy eigenvalues and eigenvectors? ( [2] ) (e) What are the energy eigenvalues and eigenvectors for the case of a particle with angular momentum ( j=frac{1}{2} ) ? [1]

          Consider a particle with angular momentum ( j=1 ).
(a) Write down the three basis states which are eigenvectors of ( J_{z} ) in both Dirac notation and as vectors.
[1]
(b) Write the matrices for the operators ( J^{2}, J_{+}, J_{-}, J_{x}, J_{y} ), and ( J_{z} ). Which of these operators are Hermitian?
( [15] )
(c) Which operators in part ( (b) ) form a complete set of commuting observables?
[1]
(d) Consider the Hamiltonian ( H=alpha J_{z} ) ( ( alpha ) is a real number). What are the energy eigenvalues and eigenvectors?
( [2] )
(e) What are the energy eigenvalues and eigenvectors for the case of a particle with angular momentum ( j=frac{1}{2} ) ?
[1]

$Consider a particle with angular momentum ( j=1 ). (a) Write down the three basis states which are eigenvectors of ( Jz ) in both Dirac notation and as vectors. [1] (b) Write the matrices for the operators ( J^2, J+, J-, Jx, Jy ), and ( Jz ). Which of these operators are Hermitian? ( [15] ) (c) Which operators in part ( (b) ) form a complete set of commuting observables? [1] (d) Consider the Hamiltonian ( H=alpha Jz ) ( ( alpha ) is a real number). What are the energy eigenvalues and eigenvectors? ( [2] ) (e) What are the energy eigenvalues and eigenvectors for the case of a particle with angular momentum ( j=frac12 ) ? [1]$

Added by Foster J.

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