Quantum Two

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Angular Momentum and Rotations

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Angular Momentum and RotationsIntroduction

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In classical mechanics the total angular momentum of an isolated system about any fixed point is conserved.

The existence of a conserved vector associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations.

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�⃑�=𝑟 × �⃑�

�⃑�

𝑟

�⃑�=∑ 𝑟 𝑖×𝑝𝑖

In classical mechanics the total angular momentum of an isolated system about any fixed point is conserved.

The existence of a conserved vector associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations.

6

�⃑�

�⃑�=∑ 𝑟 𝑖×𝑝𝑖

In classical mechanics the total angular momentum of an isolated system about any fixed point is conserved.

The existence of a conserved vector associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations.

7

�⃑�

That is, if the coordinates and momenta of the entire system are rotated "rigidly" about some point, the energy of the system is unchanged and, more importantly, it is the same function of the dynamical variables as it was before the rotation.

Such a circumstance would not apply, e.g., to a system lying in an externally imposed electric or gravitational field pointing in some specific direction.

Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external fields of this sort, space is isotropic; it behaves the same way in all directions.

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That is, if the coordinates and momenta of the entire system are rotated "rigidly" about some point, the energy of the system is unchanged and, more importantly, it is the same function of the dynamical variables as it was before the rotation.

Such a circumstance would not apply, e.g., to asystem lying in an externally imposed electric or gravitational field pointing in some specific direction.

Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external fields of this sort, space is isotropic; it behaves the same way in all directions.

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That is, if the coordinates and momenta of the entire system are rotated "rigidly" about some point, the energy of the system is unchanged and, more importantly, it is the same function of the dynamical variables as it was before the rotation.

Such a circumstance would not apply, e.g., to asystem lying in an externally imposed electric or gravitational field pointing in some specific direction.

Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external fields of this sort, space is isotropic; it behaves the same way in all directions.

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That is, if the coordinates and momenta of the entire system are rotated "rigidly" about some point, the energy of the system is unchanged and, more importantly, it is the same function of the dynamical variables as it was before the rotation.

Such a circumstance would not apply, e.g., to asystem lying in an externally imposed electric or gravitational field pointing in some specific direction.

Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external fields of this sort, space is isotropic; it behaves the same way in all directions.

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9-year WMAP image of cosmic background radiation (2012).

Not surprisingly, therefore, in quantum mechanics the individual Cartesian components , and of the total angular momentum operator of an isolated system are also constants of the motion.

The different components of are not, however, compatible quantum observables.

Indeed, as we will see the operators representing the components of angular momentum along different directions do not generally commute with one another.

Thus, the vector operator is not observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components).

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[𝐿𝑥 ,𝐿𝑦 ]≠0

Not surprisingly, therefore, in quantum mechanics the individual Cartesian components , and of the total angular momentum operator of an isolated system are also constants of the motion.

The different components of are not, however, compatible quantum observables.

Indeed, as we will see the operators representing the components of angular momentum along different directions do not generally commute with one another.

Thus, the vector operator is not observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components).

13

[𝐿𝑥 ,𝐿𝑦 ]≠0

Not surprisingly, therefore, in quantum mechanics the individual Cartesian components , and of the total angular momentum operator of an isolated system are also constants of the motion.

The different components of are not, however, compatible quantum observables.

Indeed, as we will see, the operators representing the components of angular momentum along different directions do not generally commute with one another.

Thus, the vector operator is not observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components).

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[𝐿𝑥 ,𝐿𝑦 ]≠0

Not surprisingly, therefore, in quantum mechanics the individual Cartesian components , and of the total angular momentum operator of an isolated system are also constants of the motion.

The different components of are not, however, compatible quantum observables.

Indeed, as we will see the operators, representing the components of angular momentum along different directions do not generally commute with one another.

Thus, the vector operator is not an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components).

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[𝐿𝑥 ,𝐿𝑦 ]≠0

The non-commutivity of these quantum observables often seems, at first encounter, a bit of a nuisance, and it seems to be part of the strangeness of quantum mechanics itself.

But, this is not true.

In fact, it intimately reflects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations, even of classical objects, in three dimensions about different axes do not commute with one another.

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The non-commutivity of these quantum observables often seems, at first encounter, a bit of a nuisance, and it seems to be part of the strangeness of quantum mechanics itself.

But, this is not true.

In fact, it intimately reflects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations, even of classical objects, in three dimensions about different axes do not commute with one another.

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The non-commutivity of these quantum observables often seems, at first encounter, a bit of a nuisance, and it seems to be part of the strangeness of quantum mechanics itself.

But, this is not true.

In fact, it intimately reflects the underlying structure of the three dimensional space in which we are immersed.It has its source in the fact that rotations, even of classical objects, in three dimensions about different axes do not commute with one another.

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The non-commutivity of these quantum observables often seems, at first encounter, a bit of a nuisance, and it seems to be part of the strangeness of quantum mechanics itself.

But, this is not true.

In fact, it intimately reflects the underlying structure of the three dimensional space in which we are immersed.It has its source in the fact that rotations, even of classical objects, in three dimensions about different axes do not commute with one another.

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The non-commutivity of these quantum observables often seems, at first encounter, a bit of a nuisance, and it seems to be part of the strangeness of quantum mechanics itself.

But, this is not true.

In fact, it intimately reflects the underlying structure of the three dimensional space in which we are immersed.It has its source in the fact that rotations, even of classical objects, in three dimensions about different axes do not commute with one another.

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Indeed, this non-commutivity imparts to angular momentum observables a rich characteristic structure and makes them quite useful, e.g., in classifying the bound states of atomic, molecular, and nuclear systems containing one or more particles

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Just as importantly, the existence of "spin" degrees of freedom, i.e., intrinsic angular momenta associated with the internal structure of fundamental particles,

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Just as importantly, the existence of "spin" degrees of freedom, i.e., intrinsic angular momenta associated with the internal structure of fundamental particles, provides additional motivation for the study of angular momentum, and to the general transformation properties exhibited by quantum systems subjected to rotations in three dimensions.

In the next segment, therefore, we begin our formal study of angular momentum observables by reviewing the definition and basic properties of the angular momentum of one or more particles.

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Just as importantly, the existence of "spin" degrees of freedom, i.e., intrinsic angular momenta associated with the internal structure of fundamental particles, provides additional motivation for the study of angular momentum, and to the general transformation properties exhibited by quantum systems subjected to rotations in three dimensions.

In the next segment, therefore, we begin our formal study of angular momentum observables by reviewing the definition and basic properties of the angular momentum of one or more particles.

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