Chapter 6 Angular Momentum Definition 6 B 1

Chapter 6 Angular Momentum

Definition 6. B. 1 • 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. :

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

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

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

Total angular momentum 6. B. 2 • Thereby, the spin can be treated as an intrinsic (nonorbital) 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 • Let us consider: • Similarly, one can obtain: • Thereby: 6. B. 2

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

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

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

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

Eigenvalues • Let us consider an eigenvalue problem: • Recalling the expression • Moreover 6. C. 1 6. C. 2

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 6. C. 1 6. C. 2

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

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

Eigenvalues • Combining: • There are two solutions: • Thereby: • j must be integer or half-integer 6. C. 2

Eigenvalues • Using: • So: • We will therefore use indices j and m to label the eigenstates common to J 2 and Jz • However, J 2 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 6. C. 2

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

Eigenstates • Let us recall: • And • Thus • Therefore • We need to calculate the a± constants 6. C. 3

Eigenstates • Since • We have • Calculating a matrix element • On the other hand 6. C. 3

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

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

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

6. C. 3 Eigenstates • A matrix representation of such basis can be built using subspaces E (k, j) as follows: E (k, j) E (k’, j’) E (k, j) matrix (2 j + 1) × (2 j + 1) 0 0 E (k’, j) 0 matrix (2 j + 1) × (2 j + 1) 0 E (k’, j’) 0 0 … 0 0 • It easy to show that: … 0 0 matrix 0 (2 j’ + 1) × (2 j’+ 1) 0

Eigenstates • Using: 6. C. 3

Eigenstates 6. C. 3 • 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 (2 j + 1) × (2 j + 1) = 3 × 3 • In this case:

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

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

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

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

Eigenproblem for orbital momentum • Then 6. D. 1

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

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

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: 6. D. 1

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

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

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

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

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

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

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

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

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

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

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

Properties of eigenstates 6. D. 2

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

Measurements • Consider a particle with a wave function: • In can be expanded as: • The probability of finding in a simultaneous measurement of L 2 and Lz values ћ 2 l(l+1) and ћm is 6. D. 2

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

Measurements • Then • And 6. D. 2