Quantum Mechanics: Spin Particles Quiz

Chapter 5: Angular Momentum

• This chapter introduces three-dimensional problems, a crucial prelude to Chapter 6, essential for understanding atomic systems.
• Angular momentum is vital in both classical and quantum mechanics, particularly for systems with spherically symmetric potentials.
• Bohr's model of the hydrogen atom uses quantized angular momentum.
• Quantum angular momentum is fundamental to understanding molecular, atomic, and nuclear systems.
• Classical angular momentum (L) is defined as the cross product of position (r) and momentum (p).
• The orbital angular momentum operator (L) is obtained by substituting position (R) and momentum (P = -iħ∇) operators into the classical expression.
• The Cartesian components of the angular momentum operator Lx, Ly, and Lz are given by corresponding equations.
• The square of the angular momentum (L²) is a conserved quantity that commutes with the Cartesian components of the angular momentum.
• Commutation relations between angular momentum components are crucial; they involve ih and one of the other components.
• Angular momentum operators do not commute with each other.
• Simultaneous measurement of non-commuting operators is not possible due to the uncertainty principle.
• The commutation relations allow the derivation of the general formalism of angular momentum, which is independent of a particular representation.
• Key example of commutators are calculated
• A general angular momentum operator (J) is defined, with commutation relations similar to those of orbital angular momentum.
• The square of the angular momentum operator (J²) is a scalar operator, commuting with its Cartesian components.
• The key is to diagonalize both the square and one of the components (often Lz).
• Eigenstates of the angular momentum operator (J²) and one component (e.g., Jz) are introduced.
• The eigenvalues of J² and Jz are directly related to the quantum number j and m, with m ranging from -j to +j, where j can be integer or half-integer.
• The eigenvalues of J² are given by ħ²j(j + 1) and
• The quantization of angular momentum determines the possible values of j and m.
• Raising and lowering operators (J+, J−) are introduced and their role in manipulating the eigenvectors is discussed.
• The expectation values of the x and y components of the angular momentum operator are zero.

General Formalism of Angular Momentum

• Commutation relations are introduced defining the angular momentum operators.
• Eigenstates and eigenvalues of the square of angular momentum and components.
• The factors ħ and the importance of orthonormality of the obtained eigenstates are stated.
• Raising and lowering operators (J+ and J−) manipulate eigenstates, affecting only the eigenvalue m while not changing the eigenvalue a.
• The quantum number m ranges from –j to +j
• The quantum number j is either an integer or a half-integer.
• The square of the angular momentum (J²) has eigenvalues ħ²j(j + 1).

Matrix Representation of Angular Momentum

• The general formalism is extended to matrix representations in a given basis of orthonormal eigenstates |j,m).
• Specific examples for j=1 are given.
• Matrix representations of operators Lx, Ly, and Lz are shown with explicit forms.
• The results for operators are derived; the form of matrices is important.

Geometrical Representation of Angular Momentum

• Angular momentum represented as a rotating vector on the surface of a cone around its z axis.

Experimental Evidence of Spin

• Stern-Gerlach experiment demonstrated electron's intrinsic angular momentum (spin).
• Spin's direction is quantized, unlike orbital angular momentum.
• Spin is an intrinsic property of a particle.

Spin Angular Momentum

• Spin has a quantum mechanical origin.
• The spin angular momentum operator (Ŝ) has properties analogous to orbital angular momentum operators
• The components (Sx, Sy, Sz) of the spin angular momentum operator follow analogous commutation relations to those for orbital angular momentum.
• Spin has eigenvalues related to a quantum number s, with s= 1/2 for an electron.
• The Stern-Gerlach experiment demonstrated the quantization of electron spin.
• Electron spin plays a key role in atomic and molecular systems.

Eigenfunctions of Orbital Angular Momentum

• The eigenfunctions of the orbital angular momentum operators are spherical harmonics(Yl,m(θ, φ)).
• The functions depend only on the angular variables θ and φ.
• The spherical harmonics are derived using differential equations.
• The orthogonality and completeness of spherical harmonics are mentioned.