Question

An operator associated with an infinitesimal rotation be written as

1+ (5o x r) V

using standard notation.

(a) Define generators L, L and L, associated with such rotations using the above Show that they are Hernitian clearly stating any assumptions you make.

(b) Explain why L, L., L, and L, cannot have simultanoons eigen states in general. Are there any exceptions to this rule? (ie, are there any eigenstates , m) of L and L, which are also eigenstates of L, and L,?)

(e) Use the Henniticity of the generators I., L. of infinitesimal ro- tations to show that they preserve the normalization of any rotated wavefunction (i.c., from (r) r(r)): hence normalizations are invariant under such rotations,

Answer 1

An operator associated with an infinitesimal rotation be written as

1+ (5o x r) V

using standard notation.

(a) Define generators L, L and L, associated with such rotations using the above Show that they are Hernitian clearly stating any assumptions you make.

(b) Explain why L, L., L, and L, cannot have simultanoons eigen states in general. Are there any exceptions to this rule? (ie, are there any eigenstates , m) of L and L, which are also eigenstates of L, and L,?)

(e) Use the Henniticity of the generators I., L. of infinitesimal ro- tations to show that they preserve the normalization of any rotated wavefunction (i.c., from (r) r(r)): hence normalizations are invariant under such rotations,