Quantum Two
Angular Momentum and Rotations
Angular Momentum and Rotations Commutation Relations for Scalar and Vector Operators
The last segment revealed that for any quantum system there exists a Hermitian vector operator 𝐽 that is essential for describing rotations of the state vector |𝜓〉 and its observables 𝑄. The results of the last segment imply that a rotation 𝐑 𝑢 𝛼 of the physical system will take an observable 𝑄 onto the observable For infinitesimal rotations 𝑈 𝑢 𝛿𝛼 , this transformation law simplifies Thus, for any operator 𝑄
The last segment revealed that for any quantum system there exists a Hermitian vector operator 𝐽 that is essential for describing rotations of the state vector |𝜓〉 and its observables 𝑄. The results of the last segment imply that a rotation 𝐑 𝑢 𝛼 of the physical system will take an observable 𝑄 onto the observable For infinitesimal rotations 𝑈 𝑢 𝛿𝛼 , this transformation law simplifies Thus, for any operator 𝑄
The last segment revealed that for any quantum system there exists a Hermitian vector operator 𝐽 that is essential for describing rotations of the state vector |𝜓〉 and its observables 𝑄. The results of the last segment imply that a rotation 𝐑 𝑢 𝛼 of the physical system will take an observable 𝑄 onto the observable For infinitesimal rotations 𝑈 𝑢 𝛿𝛼 , this transformation law simplifies Thus, for any operator 𝑄
The last segment revealed that for any quantum system there exists a Hermitian vector operator 𝐽 that is essential for describing rotations of the state vector |𝜓〉 and its observables 𝑄. The results of the last segment imply that a rotation 𝐑 𝑢 𝛼 of the physical system will take an observable 𝑄 onto the observable For infinitesimal rotations 𝑈 𝑢 𝛿𝛼 , this transformation law simplifies Thus, for any operator 𝑄
The last segment revealed that for any quantum system there exists a Hermitian vector operator 𝐽 that is essential for describing rotations of the state vector |𝜓〉 and its observables 𝑄. The results of the last segment imply that a rotation 𝐑 𝑢 𝛼 of the physical system will take an observable 𝑄 onto the observable For infinitesimal rotations 𝑈 𝑢 𝛿𝛼 , this transformation law simplifies Thus, for any operator 𝑄
Now, as in classical mechanics, it is possible and useful to classify observables of the system according to the manner in which they transform under rotations. Thus, an observable 𝑄 is referred to as a scalar with respect to rotations if 𝑄′= 𝑄 for all rotations 𝐑. This implies that or . Since all finite rotations can be formed from infinitesimal rotations about the same axis, a simpler condition follows from the just-derived relation which implies that 𝑄 is a scalar if for all components of 𝐽 .
Now, as in classical mechanics, it is possible and useful to classify observables of the system according to the manner in which they transform under rotations. Thus, an observable 𝑄 is referred to as a scalar with respect to rotations if 𝑄′= 𝑄 for all rotations 𝐑. This implies that or . Since all finite rotations can be formed from infinitesimal rotations about the same axis, a simpler condition follows from the just-derived relation which implies that 𝑄 is a scalar if for all components of 𝐽 .
Now, as in classical mechanics, it is possible and useful to classify observables of the system according to the manner in which they transform under rotations. Thus, an observable 𝑄 is referred to as a scalar with respect to rotations if 𝑄′= 𝑄 for all rotations 𝐑. This implies that or . Since all finite rotations can be formed from infinitesimal rotations about the same axis, a simpler condition follows from the just-derived relation which implies that 𝑄 is a scalar if for all components of 𝐽 .
Now, as in classical mechanics, it is possible and useful to classify observables of the system according to the manner in which they transform under rotations. Thus, an observable 𝑄 is referred to as a scalar with respect to rotations if 𝑄′= 𝑄 for all rotations 𝐑. This implies that or . Since all finite rotations can be formed from infinitesimal rotations about the same axis, a simpler condition follows from the just-derived relation which implies that 𝑄 is a scalar if for all components of 𝐽 .
Now, as in classical mechanics, it is possible and useful to classify observables of the system according to the manner in which they transform under rotations. Thus, an observable 𝑄 is referred to as a scalar with respect to rotations if 𝑄′= 𝑄 for all rotations 𝐑. This implies that or . Since all finite rotations can be formed from infinitesimal rotations about the same axis, a simpler condition follows from the just-derived relation which implies that 𝑄 is a scalar if for all components of 𝐽 .
Now, as in classical mechanics, it is possible and useful to classify observables of the system according to the manner in which they transform under rotations. Thus, an observable 𝑄 is referred to as a scalar with respect to rotations if 𝑄′= 𝑄 for all rotations 𝐑. This implies that or . Since all finite rotations can be formed from infinitesimal rotations about the same axis, a simpler condition follows from the just-derived relation Thus, 𝑄 is a scalar with respect to rotations if for all 𝑢 .
Now, as in classical mechanics, it is possible and useful to classify observables of the system according to the manner in which they transform under rotations. Thus, an observable 𝑄 is referred to as a scalar with respect to rotations if 𝑄′= 𝑄 for all rotations 𝐑. This implies that or . Since all finite rotations can be formed from infinitesimal rotations about the same axis, a simpler condition follows from the just-derived relation Thus, 𝑄 is a scalar with respect to rotations if for all 𝑖. These are what we will refer to as scalar commutation relations.
Now, as in classical mechanics, it is possible and useful to classify observables of the system according to the manner in which they transform under rotations. Thus, an observable 𝑄 is referred to as a scalar with respect to rotations if 𝑄′= 𝑄 for all rotations 𝐑. This implies that or . Since all finite rotations can be formed from infinitesimal rotations about the same axis, a simpler condition follows from the just-derived relation Thus, 𝑄 is a scalar with respect to rotations if for all 𝑖. These are what we will refer to as scalar commutation relations.
Similarly, as we have seen, a collection of three operators 𝑉 𝑥 , 𝑉 𝑦 , and 𝑉 𝑧 can be viewed as forming the components of a vector operator 𝑉 if the component of 𝑉 along an arbitrary direction 𝑎 is 𝑉 𝑎 = 𝑉 ⋅ 𝑎 = 𝑖 𝑉 𝑖 𝑎 𝑖 . The Hermitian operator 𝐽 is a vector operator, since its component along any direction is a linear combination of its three Cartesian components with coefficients that are, indeed, just the associated direction cosines. But this does not mean that we can just choose any three operators and associate them with the components of a vector operator. It is necessary, also , that those components transform the right way under rotations.
Similarly, as we have seen, a collection of three operators 𝑉 𝑥 , 𝑉 𝑦 , and 𝑉 𝑧 can be viewed as forming the components of a vector operator 𝑉 if the component of 𝑉 along an arbitrary direction 𝑎 is 𝑉 𝑎 = 𝑉 ⋅ 𝑎 = 𝑖 𝑉 𝑖 𝑎 𝑖 . The Hermitian operator 𝐽 is a vector operator, since its component along any direction is a linear combination of its three Cartesian components with coefficients that are, indeed, just the associated direction cosines. But this does not mean that we can just choose any three operators and associate them with the components of a vector operator. It is necessary, also , that those components transform the right way under rotations.
Similarly, as we have seen, a collection of three operators 𝑉 𝑥 , 𝑉 𝑦 , and 𝑉 𝑧 can be viewed as forming the components of a vector operator 𝑉 if the component of 𝑉 along an arbitrary direction 𝑎 is 𝑉 𝑎 = 𝑉 ⋅ 𝑎 = 𝑖 𝑉 𝑖 𝑎 𝑖 . The Hermitian operator 𝐽 is a vector operator, since its component along any direction is a linear combination of its three Cartesian components with coefficients that are, indeed, just the associated direction cosines. But this does not mean that we can just choose any three operators and associate them with the components of a vector operator. It is necessary, also , that those components transform the right way under rotations.
Similarly, as we have seen, a collection of three operators 𝑉 𝑥 , 𝑉 𝑦 , and 𝑉 𝑧 can be viewed as forming the components of a vector operator 𝑉 if the component of 𝑉 along an arbitrary direction 𝑎 is 𝑉 𝑎 = 𝑉 ⋅ 𝑎 = 𝑖 𝑉 𝑖 𝑎 𝑖 . The Hermitian operator 𝐽 is a vector operator, since its component along any direction is a linear combination of its three Cartesian components with coefficients that are, indeed, just the associated direction cosines. But this does not mean that we can just choose any three operators and associate them with the components of a vector operator. It is necessary, also , that those components transform the right way under rotations.
Now, after undergoing a rotation 𝐑, a device initially setup to measure the component 𝑉 𝑎 of a vector operator 𝑉 along the direction 𝑎 will now measure the component of 𝑉 along the rotated direction 𝑎 ′ = 𝐴 𝐑 𝑎 where 𝐴 𝐑 is the orthogonal matrix associated with the rotation 𝐑. Thus we can write 𝑢 |𝜓〉 𝑉 𝑎
Now, after undergoing a rotation 𝐑, a device initially setup to measure the component 𝑉 𝑎 of a vector operator 𝑉 along the direction 𝑎 will now measure the component of 𝑉 along the rotated direction 𝑎 ′ = 𝐴 𝐑 𝑎 where 𝐴 𝐑 is the orthogonal matrix associated with the rotation 𝐑. Thus we can write 𝑢 |𝜓′〉 𝑉 𝑎 ′
Now, after undergoing a rotation 𝐑, a device initially setup to measure the component 𝑉 𝑎 of a vector operator 𝑉 along the direction 𝑎 will now measure the component of 𝑉 along the rotated direction 𝑎 ′ = 𝐴 𝐑 𝑎 where 𝐴 𝐑 is the orthogonal matrix associated with the rotation 𝐑. Thus we can write 𝑢 |𝜓′〉 𝑉 𝑎 ′
Now, after undergoing a rotation 𝐑, a device initially setup to measure the component 𝑉 𝑎 of a vector operator 𝑉 along the direction 𝑎 will now measure the component of 𝑉 along the rotated direction 𝑎 ′ = 𝐴 𝐑 𝑎 where 𝐴 𝐑 is the orthogonal matrix associated with the rotation 𝐑. Thus we can write 𝑢 |𝜓′〉 𝑉 𝑎 ′
Now, after undergoing a rotation 𝐑, a device initially setup to measure the component 𝑉 𝑎 of a vector operator 𝑉 along the direction 𝑎 will now measure the component of 𝑉 along the rotated direction 𝑎 ′ = 𝐴 𝐑 𝑎 where 𝐴 𝐑 is the orthogonal matrix associated with the rotation 𝐑. Thus we can write 𝑢 |𝜓′〉 𝑉 𝑎 ′
Now, after undergoing a rotation 𝐑, a device initially setup to measure the component 𝑉 𝑎 of a vector operator 𝑉 along the direction 𝑎 will now measure the component of 𝑉 along the rotated direction 𝑎 ′ = 𝐴 𝐑 𝑎 where 𝐴 𝐑 is the orthogonal matrix associated with the rotation 𝐑. Thus we can write 𝑢 |𝜓′〉 𝑉 𝑎 ′
Now, after undergoing a rotation 𝐑, a device initially setup to measure the component 𝑉 𝑎 of a vector operator 𝑉 along the direction 𝑎 will now measure the component of 𝑉 along the rotated direction 𝑎 ′ = 𝐴 𝐑 𝑎 where 𝐴 𝐑 is the orthogonal matrix associated with the rotation 𝐑. Thus we can write 𝑢 |𝜓′〉 𝑉 𝑎 ′
Again focusing on infinitesimal rotations 𝑈 𝑢 𝛿𝛼 , and applying our previously derived relation with 𝑄′= 𝑉 ⋅ 𝑎 ′, 𝑄= 𝑉 ⋅ 𝑎 , and writing 𝐽 𝑢 = 𝐽 ⋅ 𝑢 gives the relation But as we have shown, an infinitesimal rotation 𝐴 𝑢 𝛿𝛼 about 𝑢 takes a vector 𝑎 onto the vector so that, just by geometry comparing these expressions, we find that
Again focusing on infinitesimal rotations 𝑈 𝑢 𝛿𝛼 , and applying our previously derived relation with 𝑄′= 𝑉 ⋅ 𝑎 ′, 𝑄= 𝑉 ⋅ 𝑎 , and writing 𝐽 𝑢 = 𝐽 ⋅ 𝑢 gives the relation But as we have shown, an infinitesimal rotation 𝐴 𝑢 𝛿𝛼 about 𝑢 takes a vector 𝑎 onto the vector so that, just by geometry comparing these expressions, we find that
Again focusing on infinitesimal rotations 𝑈 𝑢 𝛿𝛼 , and applying our previously derived relation with 𝑄′= 𝑉 ⋅ 𝑎 ′, 𝑄= 𝑉 ⋅ 𝑎 , and writing 𝐽 𝑢 = 𝐽 ⋅ 𝑢 gives the relation But as we have shown, an infinitesimal rotation 𝐴 𝑢 𝛿𝛼 about 𝑢 takes a vector 𝑎 onto the vector so that, just by geometry comparing these expressions, we find that
Again focusing on infinitesimal rotations 𝑈 𝑢 𝛿𝛼 , and applying our previously derived relation with 𝑄′= 𝑉 ⋅ 𝑎 ′, 𝑄= 𝑉 ⋅ 𝑎 , and writing 𝐽 𝑢 = 𝐽 ⋅ 𝑢 gives the relation But as we have shown, an infinitesimal rotation 𝐴 𝑢 𝛿𝛼 about 𝑢 takes a vector 𝑎 onto the vector so that, just by geometry comparing these expressions, we find that
Again focusing on infinitesimal rotations 𝑈 𝑢 𝛿𝛼 , and applying our previously derived relation with 𝑄′= 𝑉 ⋅ 𝑎 ′, 𝑄= 𝑉 ⋅ 𝑎 , and writing 𝐽 𝑢 = 𝐽 ⋅ 𝑢 gives the relation But as we have shown, an infinitesimal rotation 𝐴 𝑢 𝛿𝛼 about 𝑢 takes a vector 𝑎 onto the vector so that, just by geometry comparing these expressions, we find that
Canceling the common factor of 𝛿𝛼, multiplying through by I, and re-arranging things a little leads to the relation Taking 𝑢 and 𝑎 along the 𝑖th and 𝑗th Cartesian axes, respectively, this latter relation can be written in the illuminating form which we will refer to as vector commutation relations
Canceling the common factor of 𝛿𝛼, multiplying through by 𝑖, and re-arranging things a little leads to the relation Taking 𝑢 and 𝑎 along the 𝑖th and 𝑗th Cartesian axes, respectively, this latter relation can be written in the illuminating form which we will refer to as vector commutation relations
Canceling the common factor of 𝛿𝛼, multiplying through by 𝑖, and re-arranging things a little leads to the relation Taking 𝑢 and 𝑎 along the 𝑖th and 𝑗th Cartesian axes, respectively, this latter relation can be written in the illuminating form which we will refer to as vector commutation relations
Canceling the common factor of 𝛿𝛼, multiplying through by 𝑖, and re-arranging things a little leads to the relation Taking 𝑢 and 𝑎 along the 𝑖th and 𝑗th Cartesian axes, respectively, this latter relation can be written in the illuminating form which we will refer to as vector commutation relations
Canceling the common factor of 𝛿𝛼, multiplying through by 𝑖, and re-arranging things a little leads to the relation Taking 𝑢 and 𝑎 along the 𝑖th and 𝑗th Cartesian axes, respectively, this latter relation can be written in the illuminating form which we will refer to as vector commutation relations
Canceling the common factor of 𝛿𝛼, multiplying through by 𝑖, and re-arranging things a little leads to the relation Taking 𝑢 and 𝑎 along the 𝑖th and 𝑗th Cartesian axes, respectively, this latter relation can be written in the illuminating form which we will refer to as vector commutation relations.
writing these out explicitly we see that and note that the components of 𝐽 and 𝑉 along the same direction commute. This shows that the components of any vector operator 𝑉 of a quantum system obey commutation relations with the components of 𝐽 that are very similar to those derived earlier for the operator components of the orbital angular momentum.
writing these out explicitly we see that and note that the components of 𝐽 and 𝑉 along the same direction commute. This shows that the components of any vector operator 𝑉 of a quantum system obey commutation relations with the components of 𝐽 that are very similar to those derived earlier for the operator components of the orbital angular momentum.
writing these out explicitly we see that and note that the components of 𝐽 and 𝑉 along the same direction commute. This shows that the components of any vector operator 𝑉 of a quantum system obey commutation relations with the components of 𝐽 that are very similar to those derived earlier for the operator components of the orbital angular momentum.
Indeed, since the quantum system that we are considering is completely arbitrary, the only vector operator that we actually know that it possesses is the operator 𝐽 itself. Thus, the components of 𝐽 obey vector commutation relations, i.e., which as we have already seen then implies that So for 𝐽 itself, vector commutation relations take the special form of angular momentum commutation relations.
Indeed, since the quantum system that we are considering is completely arbitrary, the only vector operator that we actually know that it possesses is the operator 𝐽 itself. Thus, the components of 𝐽 obey vector commutation relations, i.e., which as we have already seen then implies that So for 𝐽 itself, vector commutation relations take the special form of angular momentum commutation relations.
Indeed, since the quantum system that we are considering is completely arbitrary, the only vector operator that we actually know that it possesses is the operator 𝐽 itself. Thus, the components of 𝐽 obey vector commutation relations, i.e., which as we have already seen then implies that So for 𝐽 itself, vector commutation relations take the special form of angular momentum commutation relations.
Indeed, since the quantum system that we are considering is completely arbitrary, the only vector operator that we actually know that it possesses is the operator 𝐽 itself. Thus, the components of 𝐽 obey vector commutation relations, i.e., which as we have already seen then implies that So for 𝐽 itself, vector commutation relations take the special form of angular momentum commutation relations.
Indeed, since the quantum system that we are considering is completely arbitrary, the only vector operator that we actually know that it possesses is the operator 𝐽 itself. Thus, the components of 𝐽 obey vector commutation relations, i.e., which as we have already seen then implies that So for 𝐽 itself, vector commutation relations take the special form of angular momentum commutation relations.
Indeed, since the quantum system that we are considering is completely arbitrary, the only vector operator that we actually know that it possesses is the operator 𝐽 itself. Thus, the components of 𝐽 obey vector commutation relations, i.e., which as we have already seen then implies that So for 𝐽 itself, vector commutation relations take the special form of angular momentum commutation relations.
Thus, our plan of identifying the operator 𝐽 , whose components give the generators of rotation, with the total angular momentum of the quantum system is entirely on track. Indeed, the operator 𝐽 is certainly an angular momentum since it obey the characteristic angular momentum commutation relations. In the next segment, we show that for a single quantum mechanical particle, we can directly identify 𝐽 = ℓ = 𝑅 × 𝐾
Thus, our plan of identifying the operator 𝐽 , whose components give the generators of rotation, with the total angular momentum of the quantum system is entirely on track. Indeed, the operator 𝐽 is certainly an angular momentum since it obey the characteristic angular momentum commutation relations. In the next segment, we show that for a single quantum mechanical particle, we can directly identify 𝐽 = ℓ = 𝑅 × 𝐾
Thus, our plan of identifying the operator 𝐽 , whose components give the generators of rotation, with the total angular momentum of the quantum system is entirely on track. Indeed, the operator 𝐽 is certainly an angular momentum since it obey the characteristic angular momentum commutation relations. In the next segment, we show that for a single quantum mechanical particle, we can, in fact, directly identify the generator of rotations 𝐽 = ℓ = 𝑅 × 𝐾 with the particle’s orbital angular momentum.