Chapter 6 Angular Momentum

6.B.1 Definition In classical mechanics the angular momentum of a point mass relative to some axis is defined as: In quantum mechanics the orbital angular momentum is defined by applying a quantization operation to the classical expression – replacing classical physical quantities with the corresponding observables (operators): This definition does not require symmetrization with respect to non-commuting operators, e.g.:

6.B.1 Definition For a system of (spinless) particles the total orbital angular momentum is defined as:

Commutation relations 6.B.1 Commutation relations Let us consider: Similarly, one can obtain: Thereby:

Total angular momentum 6.B.2 Total angular momentum Previously we introduced the spin: Let us consider: Similarly, one can obtain:

Total angular momentum 6.B.2 Total angular momentum Thereby, the spin can be treated as an intrinsic (non-orbital) angular momentum Combination of the orbital and intrinsic angular momenta is the total angular momentum of the system J, defined via the commutation relations: Let us introduce an operator: This operator is Hermitian, and we will assume it is an observable

Total angular momentum 6.B.2 Total angular momentum Let us consider: Similarly, one can obtain: Thereby:

Total angular momentum 6.B.2 Total angular momentum What is the physical meaning of the commutation relations? It is impossible to measure simultaneously the three components of the angular momentum, however, J2 and any component of J are compatible and could be measured simultaneously Therefore, there is a possibility to find simultaneous eigenstates of J2 and any component of J (e.g., Jz)

Total angular momentum 6.C.1 Total angular momentum We introduce (non-Hermitian) operators: Let us consider: Also

Total angular momentum 6.C.1 Total angular momentum We introduce (non-Hermitian) operators: Let us consider: Synopsizing:

Total angular momentum 6.C.1 Total angular momentum Let us also calculate: Similarly: Therefore:

Eigenvalues Let us consider an eigenvalue problem: Recalling the expression Moreover

Eigenvalues Let us consider an eigenvalue problem: This also can be written as: And this also can be written as:

Eigenvalues Let us consider an eigenvalue problem: This also can be written as: And this also can be written as:

6.C.1 6.C.2 Eigenvalues Since: On the other hand, the square of the eigenvalue of the z-component of the angular momentum cannot exceed the eigenvalue of its magnitude squared: Therefore, there should be top and bottom “rungs” for integers m and n

Eigenvalues Let us assume that for the top “rung” the eigenstate is: 6.C.2 Eigenvalues Let us assume that for the top “rung” the eigenstate is: Then: And: Recall:

6.C.2 Eigenvalues Let us assume that for the bottom “rung” the eigenstate is: Then: And: Recall:

Eigenvalues Combining: There are two solutions: Thereby: j must be integer or half-integer

Eigenvalues Using: So: 6.C.2 Eigenvalues Using: So: We will therefore use indices j and m to label the eigenstates common to J2 and Jz However, J2 and Jz do not necessarily constitute a CSCO, thus we introduce a third index k to distinguish between different eigenstates corresponding to the same j and m

Eigenvalues Synopsizing: 6.C.2 Eigenvalues Synopsizing: We thus have found the eigenvalues of the angular momentum What are the eigenstates?

Eigenstates Let us recall: And Thus Therefore We need to calculate the a± constants

Eigenstates Since We have Calculating a matrix element On the other hand

Eigenstates Since We have Calculating a matrix element On the other hand

Eigenstates Since We have Calculating a matrix element On the other hand

Eigenstates Since We have Calculating a matrix element On the other hand

Eigenstates Therefore A possible solution: 6.C.3 Eigenstates Therefore A possible solution: Calculating a matrix element On the other hand

Eigenstates Therefore A possible solution: 6.C.3 Eigenstates Therefore A possible solution: Assuming a normalized set of eigenstates Similarly

Eigenstates We assumed that the considered eigenstates are normalized In fact, it is practical to work with a complete orthonormal basis Thus, we should require: And the closure relation:

6.C.3 Eigenstates A matrix representation of such basis can be built using subspaces E (k,j) as follows: It easy to show that:   E (k,j) E (k’,j) E (k’,j’) … matrix (2j + 1) × (2j + 1) (2j’ + 1) × (2j’+ 1)

6.C.3 Eigenstates A matrix representation of such basis can be built using subspaces E (k,j) as follows: It easy to show that:   E (k,j) E (k’,j) E (k’,j’) … matrix (2j + 1) × (2j + 1) (2j’ + 1) × (2j’+ 1)

6.C.3 Eigenstates Using:

6.C.3 Eigenstates Using:

6.C.3 Eigenstates Matrices corresponding to subspaces E (k,j) depend on the value of j, which is determined by the specificity of the studied system For example, when j = 1, the dimensionality of the matrices is (2j + 1) × (2j + 1) = 3 × 3 In this case:

6.C.3 Eigenstates Using the expression for matrix elements below:

Eigenstates Using the expression for matrix elements below: Using 6.C.3 Eigenstates Using the expression for matrix elements below: Using

Eigenstates Using the expression for matrix elements below: Using 6.C.3 Eigenstates Using the expression for matrix elements below: Using

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum We return to the orbital angular momentum of a spinless particle Let us find relevant eigenstates in the r-representation The Cartesian components of the orbital angular momentum operator:

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum It is convenient to work in spherical coordinates

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum Then

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum Then And

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum Recall: For the orbital angular momentum: In the r-representation (and spherical coordinates):

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum In these equations r does not appear in the differential operators, so we will consider it as a parameter Thus, the wavefunction can be written as: We try separating the variables:

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum In these equations r does not appear in the differential operators, so we will consider it as a parameter Thus, the wavefunction can be written as: We try separating the variables:

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum In these equations r does not appear in the differential operators, so we will consider it as a parameter Thus, the wavefunction can be written as: We try separating the variables:

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum Then:

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum Then:

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum Then:

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum Then: Since m is integer, l shoud be also an integer (not a half-integer)

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum We successfully separated variables but still need to solve this equation: Solutions: Here Pl are the Legendre polynomials: Adrien-Marie Legendre (1752 –1833)

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum Legendre polynomials: Adrien-Marie Legendre (1752 –1833)

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum The resulting solutions: They have to be normalized This yields: Functions Y are called spherical harmonics

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum Spherical harmonics:

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum Spherical harmonics:

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum These harmonics constitute an orthonormal basis: Any function of θ and ϕ can be expanded in terms of the spherical harmonics:

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum Spherical harmonics are functions with a definite parity:

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum The original eigenproblem: In the r-representation it looks like: Where

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum This equation: In the r-representation becomes: Since One can write:

Eigenproblem for orbital momentum 6.D.1 Eigenproblem for orbital momentum Therefore: Orthonormalization relation: In the r-prerestnation it yields: Since

Properties of eigenstates 6.D.2 Properties of eigenstates Let us calculate this average value Similarly:

Properties of eigenstates 6.D.2 Properties of eigenstates Since Then: Therefore

Properties of eigenstates 6.D.2 Properties of eigenstates

Properties of eigenstates 6.D.2 Properties of eigenstates Similarly: Therefore

Measurements Consider a particle with a wave function: In can be expanded as: The probability of finding in a simultaneous measurement of L2 and Lz values ћ2l(l+1) and ћm is

6.D.2 Measurements The probability of measurement of L2 with the value ћ2l(l+1) is The probability of measurement of Lz with the value ћm is It’s more convenient to use the following expansion: Where

6.D.2 Measurements Then And