1. Quantum Two

3. Angular Momentum and Rotations

4. Angular Momentum and Rotations
Eigenstates and Eigenvalues of Angular Momentum Operators

5. Having explored the relationship between rotations and angular momenta, we now undertake a systematic study of the eigenstates and eigenvalues of a vector operator J obeying angular momentum commutation relations of the type that we have derived.
As we will see, the process for obtaining this information is very similar to that used to determine the spectrum of the eigenstates of the harmonic oscillator.
We consider, therefore, an arbitrary angular momentum operator J whose components satisfy the relations
We then note, as we did for the orbital angular momentum ℓ that, since the components do not commute with one another, cannot possess an ONB of eigenstates, i.e., states which are simultaneous eigenstates of all three of its operator components.

6. Having explored the relationship between rotations and angular momenta, we now undertake a systematic study of the eigenstates and eigenvalues of a vector operator J obeying angular momentum commutation relations of the type that we have derived.
As we will see, the process for obtaining this information is very similar to that used to determine the spectrum of the eigenstates of the harmonic oscillator.
We consider, therefore, an arbitrary angular momentum operator J whose components satisfy the relations
We then note, as we did for the orbital angular momentum ℓ that, since the components do not commute with one another, cannot possess an ONB of eigenstates, i.e., states which are simultaneous eigenstates of all three of its operator components.

7. Having explored the relationship between rotations and angular momenta, we now undertake a systematic study of the eigenstates and eigenvalues of a vector operator J obeying angular momentum commutation relations of the type that we have derived.
As we will see, the process for obtaining this information is very similar to that used to determine the spectrum of the eigenstates of the harmonic oscillator.
We consider, therefore, an arbitrary angular momentum operator J whose components satisfy the relations
We then note, as we did for the orbital angular momentum ℓ that, since the components do not commute with one another, cannot possess an ONB of eigenstates, i.e., states which are simultaneous eigenstates of all three of its operator components.

8. Having explored the relationship between rotations and angular momenta, we now undertake a systematic study of the eigenstates and eigenvalues of a vector operator J obeying angular momentum commutation relations of the type that we have derived.
As we will see, the process for obtaining this information is very similar to that used to determine the spectrum of the eigenstates of the harmonic oscillator.
We consider, therefore, an arbitrary angular momentum operator J whose components satisfy the relations
We then note, as we did for the orbital angular momentum ℓ that, since the components do not commute with one another, cannot possess an ONB of eigenstates, i.e., states which are simultaneous eigenstates of all three of its operator components.

9. In fact, one can show that the only possible eigenstates of J are those for which the angular momentum is identically zero (an 𝑠-state, in the language of spectroscopy).
Nonetheless, since, each component of commutes with , it is possible to find an ONB of eigenstates common to and to the component of J along any single chosen direction.
Usually the component of J along the 𝑧-axis is chosen, because of the simple form taken by the differential operator representing that component of orbital angular momentum in spherical coordinates.
Note, however, that due to the cyclical nature of the commutation relations, anything deduced about the spectrum and eigenstates of and must also apply to the eigenstates common to and to any other component of .

10. In fact, one can show that the only possible eigenstates of J are those for which the angular momentum is identically zero (an 𝑠-state, in the language of spectroscopy).
Nonetheless, since, each component of commutes with , it is possible to find an ONB of eigenstates common to and to the component of J along any single chosen direction.
Usually the component of J along the 𝑧-axis is chosen, because of the simple form taken by the differential operator representing that component of orbital angular momentum in spherical coordinates.
Note, however, that due to the cyclical nature of the commutation relations, anything deduced about the spectrum and eigenstates of and must also apply to the eigenstates common to and to any other component of .

11. In fact, one can show that the only possible eigenstates of J are those for which the angular momentum is identically zero (an 𝑠-state, in the language of spectroscopy).
Nonetheless, since, each component of commutes with , it is possible to find an ONB of eigenstates common to and to the component of J along any single chosen direction.
Usually the component of J along the 𝑧-axis is chosen, because of the simple form taken by the differential operator representing that component of orbital angular momentum in spherical coordinates.
Note, however, that due to the cyclical nature of the commutation relations, anything deduced about the spectrum and eigenstates of and must also apply to the eigenstates common to and to any other component of .

12. In fact, one can show that the only possible eigenstates of J are those for which the angular momentum is identically zero (an 𝑠-state, in the language of spectroscopy).
Nonetheless, since, each component of commutes with , it is possible to find an ONB of eigenstates common to and to the component of J along any single chosen direction.
Usually the component of J along the 𝑧-axis is chosen, because of the simple form taken by the differential operator representing that component of orbital angular momentum in spherical coordinates.
Note, however, that due to the cyclical nature of the commutation relations, anything deduced about the spectrum and eigenstates of and must also apply to the eigenstates common to and to any other component of .

13. We note also, that, as with , the operator is Hermitian and positive definite, and thus its eigenvalues must be greater than or equal to zero.
For the moment, we will ignore other quantum numbers and simply denote a common eigenstate of and as , where, by definition,
which shows that quantum number is the associated eigenvalue of the operator , while the quantum number labels, but is not equal to the corresponding eigenvalue of
Writing the eigenvalue of in this odd way, initially, will result in simple values for the quantum number .
Note that we can write any positive eigenvalue of for some

14. We note also, that, as with , the operator is Hermitian and positive definite, and thus its eigenvalues must be greater than or equal to zero.
For the moment, we will ignore other quantum numbers and simply denote a common eigenstate of and as , where, by definition,
which shows that quantum number is the associated eigenvalue of the operator , while the quantum number labels, but is not equal to the corresponding eigenvalue of
Writing the eigenvalue of in this odd way, initially, will result in simple values for the quantum number .
Note that we can write any positive eigenvalue of for some

15. We note also, that, as with , the operator is Hermitian and positive definite, and thus its eigenvalues must be greater than or equal to zero.
For the moment, we will ignore other quantum numbers and simply denote a common eigenstate of and as , where, by definition,
which shows that quantum number is the associated eigenvalue of the operator , while the quantum number labels, but is not equal to the corresponding eigenvalue of
Writing the eigenvalue of in this odd way, initially, will result in simple values for the quantum number .
Note that we can write any positive eigenvalue of for some

16. We note also, that, as with , the operator is Hermitian and positive definite, and thus its eigenvalues must be greater than or equal to zero.
For the moment, we will ignore other quantum numbers and simply denote a common eigenstate of and as , where, by definition,
which shows that quantum number is the associated eigenvalue of the operator , while the quantum number labels, but is not equal to the corresponding eigenvalue of
Writing the eigenvalue of in this odd way, initially, will result in simple values for the quantum number .
Note that we can write any positive eigenvalue of for some

17. We note also, that, as with , the operator is Hermitian and positive definite, and thus its eigenvalues must be greater than or equal to zero.
For the moment, we will ignore other quantum numbers and simply denote a common eigenstate of and as , where, by definition,
which shows that quantum number is the associated eigenvalue of the operator , while the quantum number labels, but is not equal to the corresponding eigenvalue of
Writing the eigenvalue of in this odd way, initially, will result in simple values for the quantum number .
Note that we can write any positive eigenvalue of for some

18. We note also, that, as with , the operator is Hermitian and positive definite, and thus its eigenvalues must be greater than or equal to zero.
For the moment, we will ignore other quantum numbers and simply denote a common eigenstate of and as , where, by definition,
which shows that quantum number is the associated eigenvalue of the operator , while the quantum number labels, but is not equal to the corresponding eigenvalue of
Writing the eigenvalue of in this odd way, initially, will result in simple values for the quantum number .
Note that we can write any positive eigenvalue of for some

19. We note also, that, as with , the operator is Hermitian and positive definite, and thus its eigenvalues must be greater than or equal to zero.
For the moment, we will ignore other quantum numbers and simply denote a common eigenstate of and as , where, by definition,
which shows that quantum number is the associated eigenvalue of the operator , while the quantum number labels, but is not equal to the corresponding eigenvalue of
Writing the eigenvalue of in this odd way, initially, will result in simple values for the quantum number .
Note that we can write any positive eigenvalue of for some

20. We note also, that, as with , the operator is Hermitian and positive definite, and thus its eigenvalues must be greater than or equal to zero.
For the moment, we will ignore other quantum numbers and simply denote a common eigenstate of and as , where, by definition,
which shows that quantum number is the associated eigenvalue of the operator , while the quantum number labels, but is not equal to the corresponding eigenvalue of
Writing the eigenvalue of in this odd way, initially, will result in simple values for the quantum number Note that we can write any positive eigenvalue of as for some

21. In the interest of brevity, we will refer to a state satisfying the eigenvalue equations as a “state of angular momentum ".
To proceed further, it is convenient to trade in the two components of along the 𝑥 and 𝑦 axes for the non-Hermitian operator
and its adjoint
Of course when needed, we can always get back to the original operators
and
Thus, in determining the spectrum and common eigenstates of and ,we will find it convenient to work with the set of operators
rather than the set

22. In the interest of brevity, we will refer to a state satisfying the eigenvalue equations as a “state of angular momentum ".
To proceed further, it is convenient to trade in the two components of along the 𝑥 and 𝑦 axes for the non-Hermitian operator
and its adjoint
Of course when needed, we can always get back to the original operators
and
Thus, in determining the spectrum and common eigenstates of and ,we will find it convenient to work with the set of operators
rather than the set

23. In the interest of brevity, we will refer to a state satisfying the eigenvalue equations as a “state of angular momentum ".
To proceed further, it is convenient to trade in the two components of along the 𝑥 and 𝑦 axes for the non-Hermitian operator
and its adjoint
Of course when needed, we can always get back to the original operators
and
Thus, in determining the spectrum and common eigenstates of and ,we will find it convenient to work with the set of operators
rather than the set

24. In the interest of brevity, we will refer to a state satisfying the eigenvalue equations as a “state of angular momentum ".
To proceed further, it is convenient to trade in the two components of along the 𝑥 and 𝑦 axes for the non-Hermitian operator
and its adjoint
Of course when needed, we can always get back to the original operators
and
Thus, in determining the spectrum and common eigenstates of and ,we will find it convenient to work with the set of operators
rather than the set

25. In the interest of brevity, we will refer to a state satisfying the eigenvalue equations as a “state of angular momentum ".
To proceed further, it is convenient to trade in the two components of along the 𝑥 and 𝑦 axes for the non-Hermitian operator
and its adjoint
Of course when needed, we can always get back to the original operators
and
Thus, in determining the spectrum and common eigenstates of and ,we will find it convenient to work with the set of operators
rather than the set

26. To solve this problem, we will need commutation relations for the operators in this new set.
We note first that , being a linear combination of and , must commute with , since each of those operators do as well.
The commutator of with is also readily established;
we find that or

27. To solve this problem, we will need commutation relations for the operators in this new set.
We note first that , being a linear combination of and , must commute with , since each of those operators do as well.
The commutator of with is also readily established;
we find that or

28. To solve this problem, we will need commutation relations for the operators in this new set.
We note first that , being a linear combination of and , must commute with , since each of those operators do as well.
The commutator of with is also readily established;
we find that or

29. To solve this problem, we will need commutation relations for the operators in this new set.
We note first that , being a linear combination of and , must commute with , since each of those operators do as well.
The commutator of with is also readily established;
we find that or

30. To solve this problem, we will need commutation relations for the operators in this new set.
We note first that , being a linear combination of and , must commute with , since each of those operators do as well.
The commutator of with is also readily established;
we find that or

31. To solve this problem, we will need commutation relations for the operators in this new set.
We note first that , being a linear combination of and , must commute with , since each of those operators do as well.
The commutator of with is also readily established;
we find that or

32. To solve this problem, we will need commutation relations for the operators in this new set.
We note first that , being a linear combination of and , must commute with , since each of those operators do as well.
The commutator of with is also readily established;
we find that or

33. To solve this problem, we will need commutation relations for the operators in this new set.
We note first that , being a linear combination of and , must commute with , since each of those operators do as well.
The commutator of with is also readily established;
we find that or

34. To solve this problem, we will need commutation relations for the operators in this new set.
We note first that , being a linear combination of and , must commute with , since each of those operators do as well.
The commutator of with is also readily established;
we find that or

35. Note that these can be written in the following useful form:
and
Similarly, the commutator of and is
Thus the commutation relations of interest take the form

36. Note that these can be written in the following useful form:
and
Similarly, the commutator of and is
Thus the commutation relations of interest take the form

37. Note that these can be written in the following useful form:
and
Similarly, the commutator of and is
Thus the commutation relations of interest take the form

38. Note that these can be written in the following useful form:
and
Similarly, the commutator of and is
Thus the commutation relations of interest take the form

39. Note that these can be written in the following useful form:
and
Similarly, the commutator of and is
Thus the commutation relations of interest take the form

40. Note that these can be written in the following useful form:
and
Similarly, the commutator of and is
Thus the commutation relations of interest take the form

41. Note that these can be written in the following useful form:
and
Similarly, the commutator of and is
Thus the commutation relations of interest take the form

42. Note that these can be written in the following useful form:
and
Similarly, the commutator of and is
Thus the commutation relations of interest take the form

43. Note that these can be written in the following useful form:
and
Similarly, the commutator of and is
Thus the commutation relations of interest take the form

44. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

45. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

46. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

47. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

48. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

49. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

50. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

51. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

52. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

53. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

54. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

55. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

56. It will also be necessary in what follows to express the operator in terms of the new "components“ rather than the old components
To this end we note that
and so
Similarly
Adding these last two results, dividing by two and adding gives the relation

57. With these relations we can now proceed to deduce allowed values in the spectrum of and .
Let be an arbitrary nonzero eigenvector of and with angular momentum , where the eigenvalues of satisfy the inequalities
with
Using this, and the commutation relations, we now prove a few theorems.
For a given value of , the eigenvalue must lie in the range
To show this, consider the vectors and whose squared norms are and

58. With these relations we can now proceed to deduce allowed values in the spectrum of and .
Let be an arbitrary nonzero eigenvector of and with angular momentum , where the eigenvalues of satisfy the inequalities
with
Using this, and the commutation relations, we now prove a few theorems.
For a given value of , the eigenvalue must lie in the range
To show this, consider the vectors and whose squared norms are and

59. With these relations we can now proceed to deduce allowed values in the spectrum of and .
Let be an arbitrary nonzero eigenvector of and with angular momentum , where the eigenvalues of satisfy the inequalities
with
Using this, and the commutation relations, we now prove a few theorems.
For a given value of , the eigenvalue must lie in the range
To show this, consider the vectors and whose squared norms are and

60. With these relations we can now proceed to deduce allowed values in the spectrum of and .
Let be an arbitrary nonzero eigenvector of and with angular momentum , where the eigenvalues of satisfy the inequalities
with
Using this, and the commutation relations, we now prove a few theorems.
For a given value of , the eigenvalue must lie in the range
To show this, consider the vectors and whose squared norms are and

61. With these relations we can now proceed to deduce allowed values in the spectrum of and .
Let be an arbitrary nonzero eigenvector of and with angular momentum , where the eigenvalues of satisfy the inequalities
with
Using this, and the commutation relations, we now prove a few theorems.
For a given value of , the eigenvalue must lie in the range
To show this, consider the vectors and whose squared norms are and

62. With these relations we can now proceed to deduce allowed values in the spectrum of and .
Let be an arbitrary nonzero eigenvector of and with angular momentum , where the eigenvalues of satisfy the inequalities
with
Using this, and the commutation relations, we now prove a few theorems.
For a given value of , the eigenvalue must lie in the range
To show this, consider the vectors and whose squared norms are and

63. With these relations we can now proceed to deduce allowed values in the spectrum of and .
Let be an arbitrary nonzero eigenvector of and with angular momentum , where the eigenvalues of satisfy the inequalities
with
Using this, and the commutation relations, we now prove a few theorems.
For a given value of , the eigenvalue must lie in the range
To show this, consider the vectors and whose squared norms are and

64. But we have already shown that
so the statement can be written
which clearly requires
With this implies that for positive , we must have
This is also clearly satisfied for negative Thus, for any such state we have the upper bound

65. But we have already shown that
so the statement can be written
which clearly requires
With this implies that for positive , we must have
This is also clearly satisfied for negative Thus, for any such state we have the upper bound

66. But we have already shown that
so the statement can be written
which clearly requires
With this implies that for positive , we must have
This is also clearly satisfied for negative Thus, for any such state we have the upper bound

67. But we have already shown that
so the statement can be written
which clearly requires
With this implies that for positive , we must have
This is also clearly satisfied for negative Thus, for any such state we have the upper bound

68. But we have already shown that
so the statement can be written
which clearly requires
With this implies that for positive , we must have
This is also clearly satisfied for negative Thus, for any such state we have the upper bound

69. But we have already shown that
so the statement can be written
which clearly requires
With this implies that for positive , we must have
This is also clearly satisfied for negative Thus, for any such state we have the upper bound

70. But we have already shown that
so the statement can be written
which clearly requires
With this implies that for positive , we must have
This is also clearly satisfied for negative Thus, for any such state we have the upper bound

71. But we have already shown that
so the statement can be written
which clearly requires
With this implies that for positive , we must have
This is also clearly satisfied for negative Thus, for any such state we have the upper bound

72. Similarly, we have shown that
so the statement can be written
which clearly requires
With and negative , introduce the positive quantity so that
which then requires , Multiplying this last inequality by -1 reverses that inequality and gives the lower bound
which is also obviously satisfied for any positive value of

73. Similarly, we have shown that
so the statement can be written
which clearly requires
With and negative , introduce the positive quantity so that
which then requires , Multiplying this last inequality by -1 reverses that inequality and gives the lower bound
which is also obviously satisfied for any positive value of

74. Similarly, we have shown that
so the statement can be written
which clearly requires
With and negative , introduce the positive quantity so that
which then requires , Multiplying this last inequality by -1 reverses that inequality and gives the lower bound
which is also obviously satisfied for any positive value of

75. Similarly, we have shown that
so the statement can be written
which clearly requires
With and negative , introduce the positive quantity so that
which then requires , Multiplying this last inequality by -1 reverses that inequality and gives the lower bound
which is also obviously satisfied for any positive value of

76. Similarly, we have shown that
so the statement can be written
which clearly requires
With and negative , introduce the positive quantity so that
which then requires , Multiplying this last inequality by -1 reverses that inequality and gives the lower bound
which is also obviously satisfied for any positive value of

77. Similarly, we have shown that
so the statement can be written
which clearly requires
With and negative , introduce the positive quantity so that
which then requires , Multiplying this last inequality by -1 reverses that inequality and gives the lower bound
which is also obviously satisfied for any positive value of

78. Similarly, we have shown that
so the statement can be written
which clearly requires
With and negative , introduce the positive quantity so that
which then requires , Multiplying this last inequality by -1 reverses that inequality and gives the lower bound
which is also obviously satisfied for any positive value of

79. Similarly, we have shown that
so the statement can be written
which clearly requires
With and negative , introduce the positive quantity so that
which then requires , Multiplying this last inequality by -1 reverses that inequality and gives the lower bound
which is also obviously satisfied for any positive value of

80. Similarly, we have shown that
so the statement can be written
which clearly requires
With and negative , introduce the positive quantity so that
which then requires , Multiplying this last inequality by -1 reverses that inequality and gives the lower bound
which is also obviously satisfied for any positive value of

81. Similarly, we have shown that
so the statement can be written
which clearly requires
With and negative , introduce the positive quantity so that
which then requires , Multiplying this last inequality by -1 reverses that inequality and gives the lower bound
which is also obviously satisfied for any positive value of

82. Similarly, we have shown that
so the statement can be written
which clearly requires
With and negative , introduce the positive quantity so that
which then requires , Multiplying this last inequality by -1 reverses that inequality and gives the lower bound
which is also obviously satisfied for any positive value of

83. Combining the upper and lower bounds obtained in this way, we verify that for a given value of , any state of angular momentum has a value of satisfying
Having narrowed the range for the eigenvalues of , we now prove a second theorem.
2. The vector vanishes if and only if Otherwise, is an eigenvector of and with angular momentum , i.e., it is an eigenvector of with the same eigenvalue , but it is an eigenvector of with eigenvalue that is greater by one relative to the state

84. Combining the upper and lower bounds obtained in this way, we verify that for a given value of , any state of angular momentum has a value of satisfying
Having narrowed the range for the eigenvalues of , we now prove a second theorem.
2. The vector vanishes if and only if Otherwise, is an eigenvector of and with angular momentum , i.e., it is an eigenvector of with the same eigenvalue , but it is an eigenvector of with eigenvalue that is greater by one relative to the state

85. Combining the upper and lower bounds obtained in this way, we verify that for a given value of , any state of angular momentum has a value of satisfying
Having narrowed the range for the eigenvalues of , we now prove a second theorem.
2. The vector vanishes if and only if Otherwise, is an eigenvector of and with angular momentum , i.e., it is an eigenvector of with the same eigenvalue , but it is an eigenvector of with eigenvalue that is greater by one relative to the state

86. Combining the upper and lower bounds obtained in this way, we verify that for a given value of , any state of angular momentum has a value of satisfying
Having narrowed the range for the eigenvalues of , we now prove a second theorem.
2. The vector vanishes if and only if Otherwise, is an eigenvector of and with angular momentum , i.e., it is an eigenvector of with the same eigenvalue , but it is an eigenvector of with eigenvalue that is greater by one relative to the state

87. Combining the upper and lower bounds obtained in this way, we verify that for a given value of , any state of angular momentum has a value of satisfying
Having narrowed the range for the eigenvalues of , we now prove a second theorem.
2. The vector vanishes if and only if Otherwise, is an eigenvector of and with angular momentum , i.e., it is an eigenvector of with the same eigenvalue , but it is an eigenvector of with eigenvalue that is greater by one relative to the state

88. To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if
To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is then also an eigenvector of with eigenvalue

89. To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if
To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is then also an eigenvector of with eigenvalue

90. To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if
To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is then also an eigenvector of with eigenvalue

91. To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if
To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is then also an eigenvector of with eigenvalue

92. To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if
To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is then also an eigenvector of with eigenvalue

93. To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if
To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is then also an eigenvector of with eigenvalue

94. To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if
To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is then also an eigenvector of with eigenvalue

95. To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if
To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is then also an eigenvector of with eigenvalue

96. To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if
To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is then also an eigenvector of with eigenvalue

97. To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if
To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is then also an eigenvector of with eigenvalue

98. We then prove a third final theorem:
3. The vector vanishes if and only if Otherwise, is an eigenvector of and with angular momentum , i.e., it is an eigenvector of with the same eigenvalue , but it is an eigenvector of with eigenvalue that is lower by one relative to the state
To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if , for which

99. We then prove a third final theorem:
3. The vector vanishes if and only if Otherwise, is an eigenvector of and with angular momentum , i.e., it is an eigenvector of with the same eigenvalue , but it is an eigenvector of with eigenvalue that is lower by one relative to the state
To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if , for which

100. We then prove a third final theorem:
3. The vector vanishes if and only if Otherwise, is an eigenvector of and with angular momentum , i.e., it is an eigenvector of with the same eigenvalue , but it is an eigenvector of with eigenvalue that is lower by one relative to the state
To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if , for which

101. We then prove a third final theorem:
3. The vector vanishes if and only if Otherwise, is an eigenvector of and with angular momentum , i.e., it is an eigenvector of with the same eigenvalue , but it is an eigenvector of with eigenvalue that is lower by one relative to the state
To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if , for which

102. We then prove a third final theorem:
3. The vector vanishes if and only if Otherwise, is an eigenvector of and with angular momentum , i.e., it is an eigenvector of with the same eigenvalue , but it is an eigenvector of with eigenvalue that is lower by one relative to the state
To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if , for which

103. We then prove a third final theorem:
3. The vector vanishes if and only if Otherwise, is an eigenvector of and with angular momentum , i.e., it is an eigenvector of with the same eigenvalue , but it is an eigenvector of with eigenvalue that is lower by one relative to the state
To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if , for which

104. We then prove a third final theorem:
2. The vector vanishes if and only if Otherwise, is an eigenvector of and with angular momentum , i.e., it is an eigenvector of with the same eigenvalue , but it is an eigenvector of with eigenvalue that is lower by one relative to the state
To show the first half of the statement, we note from our previous expression that
Since , given the bounds on , it follows that vanishes if and only if , for which

105. To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is also an eigenvector of with eigenvalue
With these three theorems in hand, we now proceed to restrict even further the spectra of and

106. To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is also an eigenvector of with eigenvalue
With these three theorems in hand, we now proceed to restrict even further the spectra of and

107. To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is also an eigenvector of with eigenvalue
With these three theorems in hand, we now proceed to restrict even further the spectra of and

108. To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is also an eigenvector of with eigenvalue
With these three theorems in hand, we now proceed to restrict even further the spectra of and

109. To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is also an eigenvector of with eigenvalue
With these three theorems in hand, we now proceed to restrict even further the spectra of and

110. To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is also an eigenvector of with eigenvalue
With these three theorems in hand, we now proceed to restrict even further the spectra of and

111. To prove the second part, we first use the commutation relation in the form to write
showing that is an eigenvector of with eigenvalue ,
and then observe that, because ,
showing that is also an eigenvector of with eigenvalue
Because of their effects on the states , the operator is referred to as the raising operator, since it acts to increase the component of angular momentum along the z-axis by one unit and is referred to as the lowering operator, since it acts to decrease it by one.

112. With these three theorems in hand, we now proceed to restrict even further the spectra of and .
We note, e.g., that, given any state of angular momentum we can produce a sequence of eigenvectors of and
with eigenvalues
This sequence must terminate, or else produce eigenvectors of with eigenvalues violating the upper bound
But termination can only occur when acts on the last nonzero vector of the sequence, , with eigenvalue say, and takes it on to the null vector.
But, this only occurs if

113. With these three theorems in hand, we now proceed to restrict even further the spectra of and .
We note, e.g., that, given any state of angular momentum we can produce a sequence of eigenvectors of and
with eigenvalues
This sequence must terminate, or else produce eigenvectors of with eigenvalues violating the upper bound
But termination can only occur when acts on the last nonzero vector of the sequence, , with eigenvalue say, and takes it on to the null vector.
But, this only occurs if

114. With these three theorems in hand, we now proceed to restrict even further the spectra of and .
We note, e.g., that, given any state of angular momentum we can produce a sequence of eigenvectors of and
with eigenvalue
This sequence must terminate, or else produce eigenvectors of with eigenvalues violating the upper bound
But termination can only occur when acts on the last nonzero vector of the sequence, , with eigenvalue say, and takes it on to the null vector.
But, this only occurs if

115. With these three theorems in hand, we now proceed to restrict even further the spectra of and .
We note, e.g., that, given any state of angular momentum we can produce a sequence of eigenvectors of and
with eigenvalue
This sequence must terminate, or else produce eigenvectors of with eigenvalues violating the upper bound
But termination can only occur when acts on the last nonzero vector of the sequence, , with eigenvalue say, and takes it on to the null vector.
But, this only occurs if

116. With these three theorems in hand, we now proceed to restrict even further the spectra of and .
We note, e.g., that, given any state of angular momentum we can produce a sequence of eigenvectors of and
with eigenvalue
This sequence must terminate, or else produce eigenvectors of with eigenvalues violating the upper bound
But termination can only occur when acts on the last nonzero vector of the sequence, , with eigenvalue say, and takes it on to the null vector.
But, this only occurs if

117. With these three theorems in hand, we now proceed to restrict even further the spectra of and .
We note, e.g., that, given any state of angular momentum we can produce a sequence of eigenvectors of and
with eigenvalue
This sequence must terminate, or else produce eigenvectors of with eigenvalues violating the upper bound
But termination can only occur when acts on the last nonzero vector of the sequence, , with eigenvalue say, and takes it on to the null vector.
But, this only occurs if

118. With these three theorems in hand, we now proceed to restrict even further the spectra of and .
We note, e.g., that, given any state of angular momentum we can produce a sequence of eigenvectors of and
with eigenvalues
This sequence must terminate, or else produce eigenvectors of with eigenvalues violating the upper bound
But termination can only occur when acts on the last nonzero vector of the sequence, , with eigenvalue say, and takes it on to the null vector.
But, this only occurs if

119. With these three theorems in hand, we now proceed to restrict even further the spectra of and .
We note, e.g., that, given any state of angular momentum we can produce a sequence of eigenvectors of and
with eigenvalues
This sequence must terminate, or else produce eigenvectors of with eigenvalues violating the upper bound
But termination can only occur when acts on the last nonzero vector of the sequence, , with eigenvalue say, and takes it on to the null vector.
But, this only occurs if

120. With these three theorems in hand, we now proceed to restrict even further the spectra of and .
We note, e.g., that, given any state of angular momentum we can produce a sequence of eigenvectors of and
with eigenvalues
This sequence must terminate, or else produce eigenvectors of with eigenvalues violating the upper bound
But termination can only occur when acts on the last nonzero vector of the sequence, , with eigenvalue say, and takes it on to the null vector.
But, this only occurs if

121. Thus, there exists an integer such that
Similar arguments can be made for the sequence of eigenvectors of J² and J_{z}
with eigenvalues
To now avoid violating the lower bound , the operator must act on the last nonzero vector of the sequence, with eigenvalue to take it onto the null vector.
But this only occurs if
Thus, there exists an integer such that

122. Thus, there exists an integer such that
Similar arguments can be made for the sequence of eigenvectors of and
with eigenvalue
To now avoid violating the lower bound , the operator must act on the last nonzero vector of the sequence, with eigenvalue to take it onto the null vector.
But this only occurs if
Thus, there exists an integer such that

123. Thus, there exists an integer such that
Similar arguments can be made for the sequence of eigenvectors of and
with eigenvalues
To now avoid violating the lower bound , the operator must act on the last nonzero vector of the sequence, with eigenvalue to take it onto the null vector.
But this only occurs if
Thus, there exists an integer such that

124. Thus, there exists an integer such that
Similar arguments can be made for the sequence of eigenvectors of and
with eigenvalues
To now avoid violating the lower bound , the operator must act on the last nonzero vector of the sequence, with eigenvalue to take it onto the null vector.
But this only occurs if
Thus, there exists an integer such that

125. Thus, there exists an integer such that
Similar arguments can be made for the sequence of eigenvectors of and
with eigenvalues
To now avoid violating the lower bound , the operator must act on the last nonzero vector of the sequence, with eigenvalue to take it onto the null vector.
But this only occurs if
Thus, there exists an integer such that

126. Thus, there exists an integer such that
Similar arguments can be made for the sequence of eigenvectors of and
with eigenvalues
To now avoid violating the lower bound , the operator must act on the last nonzero vector of the sequence, with eigenvalue to take it onto the null vector.
But this only occurs if
Thus, there exists an integer such that

127. Thus, there exists an integer such that
Similar arguments can be made for the sequence of eigenvectors of and
with eigenvalues
To now avoid violating the lower bound , the operator must act on the last nonzero vector of the sequence, with eigenvalue to take it onto the null vector.
But this only occurs if
Thus, there exists an integer such that

128. Thus, there exists an integer such that
Similar arguments can be made for the sequence of eigenvectors of and
with eigenvalues
To now avoid violating the lower bound , the operator must act on the last nonzero vector of the sequence, with eigenvalue to take it onto the null vector.
But this only occurs if
Thus, there exists an integer such that

129. Adding the two relations
we deduce that there exists an integer such that or
Thus, j must be either an integer or a half-integer.
If N is an even integer, then j is itself a non-negative integer in the set
For this situation, the results of the proceeding analysis indicate that m must also be an integer and, for a given integer value of j, the values must take on each of the integer values

130. Adding the two relations
we deduce that there exists an integer such that or
Thus, j must be either an integer or a half-integer.
If N is an even integer, then j is itself a non-negative integer in the set
For this situation, the results of the proceeding analysis indicate that m must also be an integer and, for a given integer value of j, the values must take on each of the integer values

131. Adding the two relations
we deduce that there exists an integer such that or
Thus, j must be either an integer or a half-integer.
If N is an even integer, then j is itself a non-negative integer in the set
For this situation, the results of the proceeding analysis indicate that m must also be an integer and, for a given integer value of j, the values must take on each of the integer values

132. Adding the two relations
we deduce that there exists an integer such that or
Thus, j must be either an integer or a half-integer.
If N is an even integer, then j is itself a non-negative integer in the set
For this situation, the results of the proceeding analysis indicate that m must also be an integer and, for a given integer value of j, the values must take on each of the integer values

133. Adding the two relations
we deduce that there exists an integer such that or
Thus, j must be either an integer or a half-integer.
If N is an even integer, then in this case, and is said to be an integral value of angular momentum.
For this situation, the results of the proceeding analysis indicate that the values must also be an integer and take on each of the integer values

134. Adding the two relations
we deduce that there exists an integer such that or
Thus, j must be either an integer or a half-integer.
If N is an even integer, then in this case, and is said to be an integral value of angular momentum.
For this situation, the results of the proceeding analysis indicate that must also be an integer and take on each of the integer values

135. If is an odd integer, then differs from an integer by 1/2, i. e
If is an odd integer, then differs from an integer by 1/2, i.e., it is in the set and is then said to be half-integral (short for half-odd- integral).
For a given half-integral value of , the values of must then take on each of the half-odd-integer values
Thus, we have deduced the values of and that are consistent with the commutation relations.
In particular, the allowed values of that can occur are the non-negative integers and the positive half-odd-integers.
For each value of j, the eigenvectors of and always come in sets, or multi- plets, of fold mutually-orthogonal eigenvectors

136. If is an odd integer, then differs from an integer by 1/2, i. e
If is an odd integer, then differs from an integer by 1/2, i.e., it is in the set and is then said to be half-integral (short for half-odd- integral).
For a given half-integral value of , the values of must then take on each of the half-odd-integer values
Thus, we have deduced the values of and that are consistent with the commutation relations.
In particular, the allowed values of that can occur are the non-negative integers and the positive half-odd-integers.
For each value of j, the eigenvectors of and always come in sets, or multi- plets, of fold mutually-orthogonal eigenvectors

137. If is an odd integer, then differs from an integer by 1/2, i. e
If is an odd integer, then differs from an integer by 1/2, i.e., it is in the set and is then said to be half-integral (short for half-odd- integral).
For a given half-integral value of , the values of must then take on each of the half-odd-integer values
Thus, we have deduced the values of and that are consistent with the commutation relations.
In particular, the allowed values of that can occur are the non-negative integers and the positive half-odd-integers.
For each value of j, the eigenvectors of and always come in sets, or multi- plets, of fold mutually-orthogonal eigenvectors

138. If is an odd integer, then differs from an integer by 1/2, i. e
If is an odd integer, then differs from an integer by 1/2, i.e., it is in the set and is then said to be half-integral (short for half-odd- integral).
For a given half-integral value of , the values of must then take on each of the half-odd-integer values
Thus, we have deduced the values of and that are consistent with the commutation relations.
In particular, the allowed values of that can occur are the non-negative integers and half-odd-integers.
For each value of j, the eigenvectors of and always come in sets, or multi- plets, of fold mutually-orthogonal eigenvectors

139. If is an odd integer, then differs from an integer by 1/2, i. e
If is an odd integer, then differs from an integer by 1/2, i.e., it is in the set and is then said to be half-integral (short for half-odd- integral).
For a given half-integral value of , the values of must then take on each of the half-odd-integer values
Thus, we have deduced the values of and that are consistent with the commutation relations.
In particular, the allowed values of that can occur are the non-negative integers and half-odd-integers.
For each value of j, the eigenvectors of and always come in sets, or multiplets, of fold mutually-orthogonal eigenvectors

140. In any given problem involving an angular momentum it must be determined which of the allowed values of and how many multiplets (corresponding to different values of other quantum numbers) for each such value of actually occur.
All the integer values of angular momentum do, in fact, arise as we will see in our study of the orbital angular momentum of a single particle.
Half-integral values of angular momentum, on the other hand, are associated with the internal or spin degrees of freedom of particles of the class of particles referred to as fermions.
Other particles, referred to as bosons, are particles that are empirically found to have integer spins.

141. In any given problem involving an angular momentum it must be determined which of the allowed values of and how many multiplets (corresponding to different values of other quantum numbers) for each such value of actually occur.
All the integer values of angular momentum do, in fact, occur in nature as we will see in our study of the orbital angular momentum of a single particle.
Half-integral values of angular momentum, on the other hand, are associated with the internal or spin degrees of freedom of particles of the class of particles referred to as fermions.
Other particles, referred to as bosons, are particles that are empirically found to have integer spins.

142. In any given problem involving an angular momentum it must be determined which of the allowed values of and how many multiplets (corresponding to different values of other quantum numbers) for each such value of actually occur.
All the integer values of angular momentum do, in fact, occur in nature as we will see in our study of the orbital angular momentum of a single particle.
Half-integral values of angular momentum, on the other hand, are associated with the internal or spin degrees of freedom of the class of particles referred to as fermions.
Other particles, referred to as bosons, are particles that are empirically found to have integer spins.

143. In any given problem involving an angular momentum it must be determined which of the allowed values of and how many multiplets (corresponding to different values of other quantum numbers) for each such value of actually occur.
All the integer values of angular momentum do, in fact, occur in nature as we will see in our study of the orbital angular momentum of a single particle.
Half-integral values of angular momentum, on the other hand, are associated with the internal or spin degrees of freedom of the class of particles referred to as fermions.
Other particles, referred to as bosons, are particles that are empirically found to have integer spins.

144. The total angular momentum of a system of particles will generally have contributions from both orbital and spin angular momenta and can be either integral or half-integral, depending upon the number and type of particles in the system.
Thus, generally speaking, there actually exist different systems in which the possible values of j and m deduced above are actually realized.
In other words, there appear to be no super-selection rules in nature that further restrict the allowed values of angular momentum from those allowed by the fundamental commutation relations.
This is sometimes said to arise from physicist Murray Gell-Mann’s Totalitarian Principle which states that (in physics at any rate)
Everything not forbidden is compulsory.

145. The total angular momentum of a system of particles will generally have contributions from both orbital and spin angular momenta and can be either integral or half-integral, depending upon the number and type of particles in the system.
Thus, generally speaking, there actually exist different systems in which the possible values of and deduced above are actually realized.
In other words, there appear to be no super-selection rules in nature that further restrict the allowed values of angular momentum from those allowed by the fundamental commutation relations.
This is sometimes said to arise from physicist Murray Gell-Mann’s Totalitarian Principle which states that (in physics at any rate)
Everything not forbidden is compulsory.

146. The total angular momentum of a system of particles will generally have contributions from both orbital and spin angular momenta and can be either integral or half-integral, depending upon the number and type of particles in the system.
Thus, generally speaking, there actually exist different systems in which the possible values of and deduced above are actually realized.
In other words, there appear to be no super-selection rules in nature that might further restrict the values of angular momentum from those allowed by the fundamental commutation relations.
This is sometimes said to arise from physicist Murray Gell-Mann’s Totalitarian Principle which states that (in physics at any rate)
Everything not forbidden is compulsory.

147. The total angular momentum of a system of particles will generally have contributions from both orbital and spin angular momenta and can be either integral or half-integral, depending upon the number and type of particles in the system.
Thus, generally speaking, there actually exist different systems in which the possible values of and deduced above are actually realized.
In other words, there appear to be no super-selection rules in nature that might further restrict the values of angular momentum from those allowed by the fundamental commutation relations.
This is sometimes said to arise from the so-called Totalitarian Principle, first enunciated by physicist Murray Gell-Mann which states that (in physics at any rate)
Everything not forbidden is compulsory.

148. The total angular momentum of a system of particles will generally have contributions from both orbital and spin angular momenta and can be either integral or half-integral, depending upon the number and type of particles in the system.
Thus, generally speaking, there actually exist different systems in which the possible values of and deduced above are actually realized.
In other words, there appear to be no super-selection rules in nature that might further restrict the values of angular momentum from those allowed by the fundamental commutation relations.
This is sometimes said to arise from the so-called Totalitarian Principle, first enunciated by physicist Murray Gell-Mann which states that (in physics at any rate)
Everything not forbidden is compulsory.