Quantum Two 1

2

Angular Momentum and Rotations 3

Eigenstates of Orbital Angular Momentum 4

In the last segment we discussed how to construct a standard representation {| , j, m 〉 } of angular momentum eigenstates for an arbitrary quantum system. We also introduced the idea of the direct sum of orthogonal quantum mechanical state spaces, and saw how the state space S can be expressed as the direct sum of the eigenspaces of any observable. Thus, e.g., we can express the state space S as a direct sum S  S j  S j   S j   … of the eigenspaces of J². Similarly each of these eigenspaces S j can be expressed as a direct sum S j  S j,   S j,   S j,   … of a certain number of identical 2j + 1 dimensional subspaces S j, . 5

In the last segment we discussed how to construct a standard representation {| , j, m 〉 } of angular momentum eigenstates for an arbitrary quantum system. We also introduced the idea of the direct sum of orthogonal quantum mechanical state spaces, and saw how the state space S can be expressed as the direct sum of the eigenspaces of any observable. Thus, e.g., we can express the state space S as a direct sum S  S j  S j   S j   … of the eigenspaces of J². Similarly each of these eigenspaces S j can be expressed as a direct sum S j  S j,   S j,   S j,   … of a certain number of identical 2j + 1 dimensional subspaces S j, . 6

In the last segment we discussed how to construct a standard representation {| , j, m 〉 } of angular momentum eigenstates for an arbitrary quantum system. We also introduced the idea of the direct sum of orthogonal quantum mechanical state spaces, and saw how the state space S can be expressed as the direct sum of the eigenspaces of any observable. Thus, e.g., we can express the state space S as a direct sum S  S j  S j   S j   … of the eigenspaces of J². Similarly each of these eigenspaces S j can be expressed as a direct sum S j  S j,   S j,   S j,   … of a certain number of identical 2j + 1 dimensional subspaces S j, . 7

In the last segment we discussed how to construct a standard representation {| , j, m 〉 } of angular momentum eigenstates for an arbitrary quantum system. We also introduced the idea of the direct sum of orthogonal quantum mechanical state spaces, and saw how the state space S can be expressed as the direct sum of the eigenspaces of any observable. Thus, e.g., we can express the state space S as a direct sum S  S j  S j   S j   … of the eigenspaces of J². Similarly each of these eigenspaces S j can be expressed as a direct sum S j  S j,   S j,   S j,   … of a certain number of identical 2j + 1 dimensional subspaces S j, . 8

In this segment, we explore some of these ideas more concretely, by applying them to the construction of a standard representation of angular momentum eigenstates for a single spinless particle, for which the total angular momentum is just the particle’s orbital angular momentum, with In the position representation these take the form of differential operators It is convenient, however, to work in standard spherical coordinates (r, θ, ϕ ), with x  r sin θ cos ϕ y  r sin θ sin ϕ z  r cos θ, in which the (dimensionless) components of ℓ take a form that is independent of the radial variable r. 9

In this segment, we explore some of these ideas more concretely, by applying them to the construction of a standard representation of angular momentum eigenstates for a single spinless particle, for which the total angular momentum is just the particle’s orbital angular momentum, with In the position representation these take the form of differential operators It is convenient, however, to work in standard spherical coordinates (r, θ, ϕ ), with x  r sin θ cos ϕ y  r sin θ sin ϕ z  r cos θ, in which the (dimensionless) components of ℓ take a form that is independent of the radial variable r. 10

In this segment, we explore some of these ideas more concretely, by applying them to the construction of a standard representation of angular momentum eigenstates for a single spinless particle, for which the total angular momentum is just the particle’s orbital angular momentum, with In the position representation these take the form of differential operators It is convenient, however, to work in standard spherical coordinates (r, θ, ϕ ), with x  r sin θ cos ϕ y  r sin θ sin ϕ z  r cos θ, in which the (dimensionless) components of ℓ take a form that is independent of the radial variable r. 11

In this segment, we explore some of these ideas more concretely, by applying them to the construction of a standard representation of angular momentum eigenstates for a single spinless particle, for which the total angular momentum is just the particle’s orbital angular momentum, with In the position representation these take the form of differential operators It is convenient, however, to work in standard spherical coordinates (r, θ, ϕ ), with x  r sin θ cos ϕ y  r sin θ sin ϕ z  r cos θ, in which the (dimensionless) components of ℓ take a form that is independent of the radial variable r. 12

Indeed, after some rather extensive use of the chain rule it is readily found that In keeping with our general development, it is also useful to construct the raising and lowering operators, which lead after a little algebra to 13

Indeed, after some rather extensive use of the chain rule it is readily found that In keeping with our general development, it is also useful to construct the raising and lowering operators, which lead after a little algebra to 14

Indeed, after some rather extensive use of the chain rule it is readily found that In keeping with our general development, it is also useful to construct the raising and lowering operators, which lead after a little algebra to 15

Indeed, after some rather extensive use of the chain rule it is readily found that In keeping with our general development, it is also useful to construct the raising and lowering operators, which lead after a little algebra to 16

Indeed, after some rather extensive use of the chain rule it is readily found that In keeping with our general development, it is also useful to construct the raising and lowering operators, which lead after a little algebra to 17

From these it is also straightforward to construct the differential operator representing the squared length of the particle’s orbital angular momentum. Now we are interested in finding common eigenstates of and. Since the differential operators are independent of r, it suffices to consider only the angular dependence. It is clear, in other words, that the eigenfunctions of these operators can be written in the form where f (r) is any acceptable function of r and the satisfy 18

From these it is also straightforward to construct the differential operator representing the squared length of the particle’s orbital angular momentum. Now we are interested in finding common eigenstates of and. Since the differential operators are independent of r, it suffices to consider only the angular dependence. It is clear, in other words, that the eigenfunctions of these operators can be written in the form where f (r) is any acceptable function of r and the satisfy 19

From these it is also straightforward to construct the differential operator representing the squared length of the particle’s orbital angular momentum. Now we are interested in finding common eigenstates of and. Since the differential operators are independent of r, it suffices to consider only the angular dependence. It is clear, in other words, that the eigenfunctions of these operators can be written in the form where f (r) is any acceptable function of r and the satisfy 20

From these it is also straightforward to construct the differential operator representing the squared length of the particle’s orbital angular momentum. Now we are interested in finding common eigenstates of and. Since the differential operators are independent of r, it suffices to consider only the angular dependence. It is clear, in other words, that the eigenfunctions of these operators can be written in the form where f (r) is any acceptable function of r and the satisfy 21

From these it is also straightforward to construct the differential operator representing the squared length of the particle’s orbital angular momentum. Now we are interested in finding common eigenstates of and. Since the differential operators are independent of r, it suffices to consider only the angular dependence. It is clear, in other words, that the eigenfunctions of these operators can be written in the form where f (r) is any acceptable function of r and the satisfy 22

the eigenvalue equations Thus, it suffices to find the functions. It will turn out that these are just the so-called spherical harmonics. This separation of radial and angular variables can also be viewed as an implicit factorization of the state space of a single particle moving in 3D in the form so that the position eigenstates |r 〉 can be viewed as defining direct product states 23

the eigenvalue equations Thus, it suffices to find the functions. It will turn out that these are just the so-called spherical harmonics. This separation of radial and angular variables can also be viewed as an implicit factorization of the state space of a single particle moving in 3D in the form so that the position eigenstates |r 〉 can be viewed as defining direct product states 24

the eigenvalue equations Thus, it suffices to find the functions. It will turn out that these are just the so-called spherical harmonics. This separation of radial and angular variables can also be viewed as an implicit factorization of the state space of a single particle moving in 3D in the form so that the position eigenstates |r 〉 can be viewed as defining direct product states 25

the eigenvalue equations Thus, it suffices to find the functions. It will turn out that these are just the so-called spherical harmonics. This separation of radial and angular variables can also be viewed as an implicit factorization of the state space of a single particle moving in 3D in the form so that the position eigenstates |r 〉 can be viewed as defining direct product states 26

The angular part can be viewed as the state space of a particle constrained to move on the surface of the unit sphere. It is spanned by an ONB of "angular position eigenstates"  θ, ϕ〉 =  Ω 〉. In this space, an arbitrary function χ(θ, ϕ ) on the unit sphere is associated with a ket where χ(θ, ϕ )  〈 θ, ϕ | χ 〉 and the integration is over all solid angle, dΩ = sin θ dθ d ϕ. 27  θ, ϕ〉

The angular part can be viewed as the state space of a particle constrained to move on the surface of the unit sphere. It is spanned by an ONB of "angular position eigenstates"  θ, ϕ〉 =  Ω 〉. In this space, an arbitrary function χ(θ, ϕ ) on the unit sphere is associated with a ket where χ(θ, ϕ )  〈 θ, ϕ | χ 〉 and the integration is over all solid angle, dΩ = sin θ dθ d ϕ. 28  θ, ϕ〉

The angular part can be viewed as the state space of a particle constrained to move on the surface of the unit sphere. It is spanned by an ONB of "angular position eigenstates"  θ, ϕ〉 =  Ω 〉. In this space, an arbitrary function χ(θ, ϕ ) on the unit sphere is associated with a ket where χ(θ, ϕ )  〈 θ, ϕ | χ 〉 and the integration is over all solid angle, dΩ = sin θ dθ d ϕ. 29  θ, ϕ〉

The angular part can be viewed as the state space of a particle constrained to move on the surface of the unit sphere. It is spanned by an ONB of "angular position eigenstates"  θ, ϕ〉 =  Ω 〉. In this space, an arbitrary function χ(θ, ϕ ) on the unit sphere is associated with a ket where χ(θ, ϕ )  〈 θ, ϕ | χ 〉 and the integration is over all solid angle, dΩ = sin θ dθ d ϕ. 30  θ, ϕ〉

The angular part can be viewed as the state space of a particle constrained to move on the surface of the unit sphere. It is spanned by an ONB of "angular position eigenstates"  θ, ϕ〉 =  Ω 〉. In this space, an arbitrary function χ(θ, ϕ ) on the unit sphere is associated with a ket where χ(θ, ϕ )  〈 θ, ϕ | χ 〉 and the integration is over all solid angle, dΩ = sin θ dθ d ϕ. 31  θ, ϕ〉

The angular part can be viewed as the state space of a particle constrained to move on the surface of the unit sphere. It is spanned by an ONB of "angular position eigenstates"  θ, ϕ〉 =  Ω 〉. In this space, an arbitrary function χ(θ, ϕ ) on the unit sphere is associated with a ket where χ(θ, ϕ )  〈 θ, ϕ | χ 〉 and the integration is over all solid angle, dΩ = sin θ dθ d ϕ. 32  θ, ϕ〉

The states  θ, ϕ〉 form a complete set of states for this space, which due to the curved space have an unusual form of the completeness relation, The normalization of these states is also slightly different from the usual Dirac normalization. Writing leads to the identification so that 33  θ, ϕ〉

The states  θ, ϕ〉 form a complete set of states for this space, which due to the curved space have an unusual form of the completeness relation, The normalization of these states is also slightly different from the usual Dirac normalization. Writing leads to the identification so that 34  θ, ϕ〉

The states  θ, ϕ〉 form a complete set of states for this space, which due to the curved space have an unusual form of the completeness relation, The normalization of these states is also slightly different from the usual Dirac normalization. Writing leads to the identification so that 35  θ, ϕ〉

The states  θ, ϕ〉 form a complete set of states for this space, which due to the curved space have an unusual form of the completeness relation, The normalization of these states is also slightly different from the usual Dirac normalization. Writing leads to the identification so that 36  θ, ϕ〉

Clearly, the components of are operators defined on this space and so we denote by  l, m 〉 the appropriate eigenstates of the Hermitian operators and within this space By assumption, then, these states satisfy the eigenvalue equations and can be expanded in the angular position representation in the form where the functions are clearly the same as those already introduced. 37

Clearly, the components of are operators defined on this space and so we denote by  l, m 〉 the appropriate eigenstates of the Hermitian operators and within this space By assumption, then, these states satisfy the eigenvalue equations and can be expanded in the angular position representation in the form where the functions are clearly the same as those already introduced. 38

Clearly, the components of are operators defined on this space and so we denote by  l, m 〉 the appropriate eigenstates of the Hermitian operators and within this space By assumption, then, these states satisfy the eigenvalue equations and can be expanded in the angular position representation in the form where the functions are clearly the same as those already introduced. 39

Clearly, the components of are operators defined on this space and so we denote by  l, m 〉 the appropriate eigenstates of the Hermitian operators and within this space By assumption, then, these states satisfy the eigenvalue equations and can be expanded in the angular position representation in the form where the functions are clearly the same as those already introduced. 40

To obtain the states | l, m 〉 (or equivalently the functions )we proceed in three stages. 1.We determine the general ϕ -dependence of the solution from the eigenvalue equation for. 2.Then, rather than solving the second order equation for directly, we determine the general form of the solution for the states | l, l 〉 having the largest component of angular momentum along the z -axis consistent with a given value of l, and which is annihilated by the raising operator 3.Finally, we use the lowering operator to develop a general formula for constructing arbitrary eigenstates of ℓ² and. 41

To obtain the states | l, m 〉 (or equivalently the functions )we proceed in three stages. 1.We determine the general ϕ -dependence of the solution from the eigenvalue equation for. 2.Then, rather than solving the second order equation for directly, we determine the general form of the solution for the states | l, l 〉 having the largest component of angular momentum along the z -axis consistent with a given value of l, and which is annihilated by the raising operator 3.Finally, we use the lowering operator to develop a general formula for constructing arbitrary eigenstates of ℓ² and. 42

To obtain the states | l, m 〉 (or equivalently the functions )we proceed in three stages. 1.We determine the general ϕ -dependence of the solution from the eigenvalue equation for. 2.Then, rather than solving the second order equation for directly, we determine the general form of the solution for the states | l, l 〉 having the largest component of angular momentum along the z -axis consistent with a given value of l, and which is annihilated by the raising operator 3.Finally, we use the lowering operator to develop a general formula for constructing arbitrary eigenstates of ℓ² and. 43

To obtain the states | l, m 〉 (or equivalently the functions )we proceed in three stages. 1.We determine the general ϕ -dependence of the solution from the eigenvalue equation for. 2.Then, rather than solving the second order equation for directly, we determine the general form of the solution for the states | l, l 〉 having the largest component of angular momentum along the z -axis consistent with a given value of l, and which is annihilated by the raising operator 3.Finally, we use the lowering operator to develop a general formula for constructing arbitrary eigenstates of ℓ² and. 44

The ϕ -dependence of the eigenfunctions in the position representation follows from the simple form taken by the operator in this representation. Indeed, the eigenfunctions of this operator must satisfy which has the general solution Single-valuedness of the wave function in this representation imposes the requirement that which leads to the restriction m ∈ {0, ±1, ±2, ⋯ }. Thus, for orbital angular momentum only integral values of m (and therefore l ) can occur. 45

The ϕ -dependence of the eigenfunctions in the position representation follows from the simple form taken by the operator in this representation. Indeed, the eigenfunctions of this operator must satisfy which has the general solution Single-valuedness of the wave function in this representation imposes the requirement that which leads to the restriction m ∈ {0, ±1, ±2, ⋯ }. Thus, for orbital angular momentum only integral values of m (and therefore l ) can occur. 46

The ϕ -dependence of the eigenfunctions in the position representation follows from the simple form taken by the operator in this representation. Indeed, the eigenfunctions of this operator must satisfy which has the general solution Single-valuedness of the wave function in this representation imposes the requirement that which leads to the restriction m ∈ {0, ±1, ±2, ⋯ }. Thus, for orbital angular momentum only integral values of m (and therefore l ) can occur. 47

The ϕ -dependence of the eigenfunctions in the position representation follows from the simple form taken by the operator in this representation. Indeed, the eigenfunctions of this operator must satisfy which has the general solution Single-valuedness of the wave function in this representation imposes the requirement that which leads to the restriction m ∈ {0, ±1, ±2, ⋯ }. Thus, for orbital angular momentum only integral values of m (and therefore l ) can occur. 48

The ϕ -dependence of the eigenfunctions in the position representation follows from the simple form taken by the operator in this representation. Indeed, the eigenfunctions of this operator must satisfy which has the general solution Single-valuedness of the wave function in this representation imposes the requirement that which leads to the restriction m ∈ {0, ±1, ±2, ⋯ }. Thus, for orbital angular momentum only integral values of m (and therefore l ) can occur. 49

The ϕ -dependence of the eigenfunctions in the position representation follows from the simple form taken by the operator in this representation. Indeed, the eigenfunctions of this operator must satisfy which has the general solution Single-valuedness of the wave function in this representation imposes the requirement that which leads to the restriction m ∈ {0, ±1, ±2, ⋯ }. Thus, for orbital angular momentum only integral values of m (and therefore l ) can occur. 50

From this result we now proceed to determine the eigenstates |l,l 〉, as represented by the wave functions To this end, we recall the general result that any such state of maximal angular momentum along the z axis is taken by the raising operator onto the null vector, i.e., In the position representation, this takes the form Letting the ϕ –derivative act replaces that derivative by the factor il. 51

From this result we now proceed to determine the eigenstates |l,l 〉, as represented by the wave functions To this end, we recall the general result that any such state of maximal angular momentum along the z axis is taken by the raising operator onto the null vector, i.e., In the position representation, this takes the form Letting the ϕ –derivative act replaces that derivative by the factor il. 52

From this result we now proceed to determine the eigenstates |l,l 〉, as represented by the wave functions To this end, we recall the general result that any such state of maximal angular momentum along the z axis is taken by the raising operator onto the null vector, i.e., In the position representation, this takes the form Letting the ϕ –derivative act replaces that derivative by the factor il. 53

From this result we now proceed to determine the eigenstates |l,l 〉, as represented by the wave functions To this end, we recall the general result that any such state of maximal angular momentum along the z axis is taken by the raising operator onto the null vector, i.e., In the position representation, this takes the form Letting the ϕ –derivative act replaces that derivative by the factor il. 54

From this result we now proceed to determine the eigenstates |l,l 〉, as represented by the wave functions To this end, we recall the general result that any such state of maximal angular momentum along the z axis is taken by the raising operator onto the null vector, i.e., In the position representation, this takes the form Letting the ϕ –derivative act replaces that derivative by the factor il. 55

From this result we now proceed to determine the eigenstates |l,l 〉, as represented by the wave functions To this end, we recall the general result that any such state of maximal angular momentum along the z axis is taken by the raising operator onto the null vector, i.e., In the position representation, this takes the form Letting the ϕ –derivative act replaces that derivative by the factor il. 56

Getting rid of the exponential factors leaves a first order differential equation which we can separate and integrate to obtain and exponentiate to obtain, up to a multiplicative constant exactly one linearly- independent acceptable solution Up to normalization we have, therefore, for each l  0, 1, 2, ⋯, the functions 57

Getting rid of the exponential factors leaves a first order differential equation which we can separate and integrate to obtain and exponentiate to obtain, up to a multiplicative constant exactly one linearly- independent acceptable solution Up to normalization we have, therefore, for each l  0, 1, 2, ⋯, the functions 58

Getting rid of the exponential factors leaves a first order differential equation which we can separate and integrate to obtain and exponentiate to obtain, up to a multiplicative constant exactly one linearly- independent acceptable solution Up to normalization we have, therefore, for each l  0, 1, 2, ⋯, the functions 59

Getting rid of the exponential factors leaves a first order differential equation which we can separate and integrate to obtain and exponentiate to obtain, up to a multiplicative constant exactly one linearly- independent acceptable solution Up to normalization we have, therefore, for each l  0, 1, 2, ⋯, the functions 60

Getting rid of the exponential factors leaves a first order differential equation which we can separate and integrate to obtain and exponentiate to obtain, up to a multiplicative constant exactly one linearly- independent acceptable solution Up to normalization we have, therefore, for each l  0, 1, 2, ⋯, the functions 61

Normalization of the orbital angular momentum eigenstates requires that Inserting a complete set of angular position states this becomes Substituting in the function the exponential factors disappear, the ϕ - integral reduces to  and we find that where … 62

Normalization of the orbital angular momentum eigenstates requires that Inserting a complete set of angular position states this becomes Substituting in the function the exponential factors disappear, the ϕ - integral reduces to  and we find that where … 63

Normalization of the orbital angular momentum eigenstates requires that Inserting a complete set of angular position states this becomes Substituting in the function the exponential factors disappear, the ϕ - integral reduces to  and we find that where … 64

Normalization of the orbital angular momentum eigenstates requires that Inserting a complete set of angular position states this becomes Substituting in the function the exponential factors disappear, the ϕ - integral reduces to  and we find that where … 65

Normalization of the orbital angular momentum eigenstates requires that Inserting a complete set of angular position states this becomes Substituting in the function the exponential factors disappear, the ϕ - integral reduces to  and we find that where … 66

Normalization of the orbital angular momentum eigenstates requires that Inserting a complete set of angular position states this becomes Substituting in the function the exponential factors disappear, the ϕ - integral reduces to  and we find that where … 67

Normalization of the orbital angular momentum eigenstates requires that Inserting a complete set of angular position states this becomes Substituting in the function the exponential factors disappear, the ϕ - integral reduces to  and we find that where … 68

where An integration by parts allows for the development of recursion relation that relates to and which leads to the result that Traditionally the phase of is chosen so that is real and positive. This leads to the relation so that, in all its glory, we have the spherical harmonic 69

where An integration by parts allows for the development of recursion relation that relates to and which leads to the result that Traditionally the phase of is chosen so that is real and positive. This leads to the relation so that, in all its glory, we have the spherical harmonic 70

where An integration by parts allows for the development of recursion relation that relates to and which leads to the result that Traditionally the phase of is chosen so that is real and positive. This leads to the relation so that, in all its glory, we have the spherical harmonic 71

where An integration by parts allows for the development of recursion relation that relates to and which leads to the result that Traditionally the phase of is chosen so that is real and positive. This leads to the relation so that, in all its glory, we have the spherical harmonic 72

where An integration by parts allows for the development of recursion relation that relates to and which leads to the result that Traditionally the phase of is chosen so that is real and positive. This leads to the relation so that, in all its glory, we have the spherical harmonic 73

where An integration by parts allows for the development of recursion relation that relates to and which leads to the result that Traditionally the phase of is chosen so that is real and positive. This leads to the relation so that, in all its glory, we have the spherical harmonic 74

From this, the remaining spherical harmonics of the same order l can be generated through application of the lowering operator, i.e., through the relation which in the position representation, after taking the derivative with respect to ϕ that occurs in the expression for ℓ₋, takes the form Thus, in this way it is straightforward to construct the angular position space wave functions for the angular momentum eigenstates of angular momentum (l, m 〉 for any value of l and m. 75

From this, the remaining spherical harmonics of the same order l can be generated through application of the lowering operator, i.e., through the relation which in the position representation, after taking the derivative with respect to ϕ that occurs in the expression for ℓ₋, takes the form Thus, in this way it is straightforward to construct the angular position space wave functions for the angular momentum eigenstates of angular momentum (l, m 〉 for any value of l and m. 76

From this, the remaining spherical harmonics of the same order l can be generated through application of the lowering operator, i.e., through the relation which in the position representation, after taking the derivative with respect to ϕ that occurs in the expression for ℓ₋, takes the form Thus, in this way it is straightforward to construct the angular position space wave functions for the angular momentum eigenstates of angular momentum (l, m 〉 for any value of l and m. 77

Properties of the angular momentum eigenfunctions It is not our intention to provide here a complete derivation of the properties of the spherical harmonics, but rather to show how they fit into the general scheme we have developed regarding angular momentum eigenstates in general. But to round things out a bit we mention without proof a number of their useful properties. 1. Orthogonality and completeness - The states |ℓ,m 〉 form an ONB of states describing a particle moving on the unit sphere. The corresponding orthonormality relation can be expressed in terms of the spherical harmonics by inserting a complete set of states to obtain 78

Properties of the angular momentum eigenfunctions It is not our intention to provide here a complete derivation of the properties of the spherical harmonics, but rather to show how they fit into the general scheme we have developed regarding angular momentum eigenstates in general. But to round things out a bit we mention without proof a number of their useful properties. 1. Orthogonality and completeness - The states |ℓ,m 〉 form an ONB of states for the space of a particle moving on the unit sphere. The corresponding orthonormality relation can be expressed in terms of the spherical harmonics by inserting a complete set of states to obtain 79

Properties of the angular momentum eigenfunctions It is not our intention to provide here a complete derivation of the properties of the spherical harmonics, but rather to show how they fit into the general scheme we have developed regarding angular momentum eigenstates in general. But to round things out a bit we mention without proof a number of their useful properties. 1. Orthogonality and completeness - The states |ℓ,m 〉 form an ONB of states for the space of a particle moving on the unit sphere. The corresponding orthonormality relation can be expressed in terms of the spherical harmonics by inserting a complete set of states to obtain 80

Properties of the angular momentum eigenfunctions It is not our intention to provide here a complete derivation of the properties of the spherical harmonics, but rather to show how they fit into the general scheme we have developed regarding angular momentum eigenstates in general. But to round things out a bit we mention without proof a number of their useful properties. 1. Orthogonality and completeness - The states |ℓ,m 〉 form an ONB of states for the space of a particle moving on the unit sphere. The corresponding orthonormality relation can be expressed in terms of the spherical harmonics by inserting a complete set of states to obtain 81

Properties of the angular momentum eigenfunctions 1.Orthogonality and completeness – Similarly, the completeness relation can be expressed in terms of the spherical harmonics by taking its matrix elements with respect to the |θ,φ 〉 states, i.e., Which our normalization convention reduces to 82

Properties of the angular momentum eigenfunctions 1.Orthogonality and completeness – Similarly, the completeness relation can be expressed in terms of the spherical harmonics by taking its matrix elements with respect to the |θ,φ 〉 states, i.e., Which our normalization convention reduces to 83

Properties of the angular momentum eigenfunctions 1.Orthogonality and completeness – Similarly, the completeness relation can be expressed in terms of the spherical harmonics by taking its matrix elements with respect to the |θ,φ 〉 states, i.e., Which our normalization convention reduces to 84

Properties of the angular momentum eigenfunctions 1.Orthogonality and completeness – Similarly, the completeness relation can be expressed in terms of the spherical harmonics by taking its matrix elements with respect to the |θ,φ 〉 states, i.e., Which our normalization convention reduces to 85

Properties of the angular momentum eigenfunctions 1.Orthogonality and completeness – Similarly, the completeness relation can be expressed in terms of the spherical harmonics by taking its matrix elements with respect to the |θ,φ 〉 states, i.e., Which our normalization convention reduces to 86

Properties of the angular momentum eigenfunctions 2.Parity - The parity operator Π acts on the eigenstates of the position representation and inverts them through the origin, i.e., Π|r 〉 =|-r 〉. It is straightforward to show that in the position representation this takes the form Πψ(r)=ψ(-r). In spherical coordinates it is also easily verified that under the parity operation and φ→φ+π. Thus, for functions on the unit sphere, Πf(θ,φ)=f(π-θ,φ+π). The parity operator commutes with the components of ℓ and with ℓ². 87

Properties of the angular momentum eigenfunctions 2.Parity - The parity operator Π acts on the eigenstates of the position representation and inverts them through the origin, i.e., Π|r 〉 =|-r 〉. It is straightforward to show that in the position representation this takes the form Πψ(r)=ψ(-r). In spherical coordinates it is also easily verified that under the parity operation and φ→φ+π. Thus, for functions on the unit sphere, Πf(θ,φ)=f(π-θ,φ+π). The parity operator commutes with the components of ℓ and with ℓ². 88

Properties of the angular momentum eigenfunctions 2.Parity - The parity operator Π acts on the eigenstates of the position representation and inverts them through the origin, i.e., Π|r 〉 =|-r 〉. It is straightforward to show that in the position representation this takes the form Πψ(r)=ψ(-r). In spherical coordinates it is also easily verified that under the parity operation and φ→φ+π. Thus, for functions on the unit sphere, Πf(θ,φ)=f(π-θ,φ+π). The parity operator commutes with the components of ℓ and with ℓ². 89

Properties of the angular momentum eigenfunctions 2.Parity - The parity operator Π acts on the eigenstates of the position representation and inverts them through the origin, i.e., Π|r 〉 =|-r 〉. It is straightforward to show that in the position representation this takes the form Πψ(r)=ψ(-r). In spherical coordinates it is also easily verified that under the parity operation and φ→φ+π. Thus, for functions on the unit sphere, Πf(θ,φ)=f(π-θ,φ+π). The parity operator commutes with the components of ℓ and with ℓ². 90

Properties of the angular momentum eigenfunctions 2.Parity - The parity operator Π acts on the eigenstates of the position representation and inverts them through the origin, i.e., Π|r 〉 =|-r 〉. It is straightforward to show that in the position representation this takes the form Πψ(r)=ψ(-r). In spherical coordinates it is also easily verified that under the parity operation and φ→φ+π. Thus, for functions on the unit sphere, Πf(θ,φ)=f(π-θ,φ+π). The parity operator commutes with the components of ℓ and with ℓ². 91

Properties of the angular momentum eigenfunctions 2.Parity - The parity operator Π acts on the eigenstates of the position representation and inverts them through the origin, i.e., Π|r 〉 =|-r 〉. It is straightforward to show that in the position representation this takes the form Πψ(r)=ψ(-r). In spherical coordinates it is also easily verified that under the parity operation and φ→φ+π. Thus, for functions on the unit sphere, Πf(θ,φ)=f(π-θ,φ+π). The parity operator commutes with the components of ℓ and with ℓ². 92

Properties of the angular momentum eigenfunctions 2.Parity - As a result, the states are eigenstates of parity and satisfy the eigenvalue equation which implies for the spherical harmonics that 93

Properties of the angular momentum eigenfunctions 2.Parity - As a result, the states are eigenstates of parity and satisfy the eigenvalue equation which implies for the spherical harmonics that 94

Properties of the angular momentum eigenfunctions 2.Parity - As a result, the states are eigenstates of parity and satisfy the eigenvalue equation which implies for the spherical harmonics that 95

Properties of the angular momentum eigenfunctions 2.Complex Conjugation - It is straightforward to show that This allows spherical harmonics with m 0. 96

Properties of the angular momentum eigenfunctions 4.Relation to the Legendre Functions - The spherical harmonics with m = 0 are directly related to the Legendre polynomials through the relation 97

Properties of the angular momentum eigenfunctions 4.Relation to the Legendre Functions - The spherical harmonics with m = 0 are directly related to the Legendre polynomials through the relation 98

Properties of the angular momentum eigenfunctions 4.Relation to the Legendre Functions - The other spherical harmonics with m > 0 are related to the associated Legendre functions through the relation 99

Properties of the angular momentum eigenfunctions 4.Relation to the Legendre Functions - The other spherical harmonics with m > 0 are related to the associated Legendre functions through the relation 100

That completes our study of the eigenstates of the angular momentum operator. In the next segment we consider implications and consequences of the notion of rotational invariance, as it applies to states, observables, and to rotationally invariant subspaces of the 3DRG. 101

That completes our study of the eigenstates of the angular momentum operator. In the next segment we consider implications and consequences of the notion of rotational invariance, as it applies to states, observables, and to rotationally invariant subspaces of the 3DRG. 102

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