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Angular Momentum What was Angular Momentum Again? • If a particle is confined to going around a sphere: At any instant the particle is on a particular circle r The particle is some distance from the origin, r The particle has angular momentum, L = r × p v The particle or mass m has some velocity, v and momentum p What was Angular Momentum Again? • So a particle going around in a circle (at any instant) has angular momentum L: L=r×p Determine L’s direction from the “right hand rule” p r What was Angular Momentum Again? • L like any 3D vector has 3 components: • Lx : projection of L on a x-axis • Ly : projection of L on a y-axis • Lz : projection of L on a z-axis L=r×p z p x r y What was Angular Momentum Again? • Picking up L and moving it over to the origin: L=r×p L=r×p z p x r y What was Angular Momentum Again? • Picking up L and moving it over to the origin: L=r×p z Rotate x y What was Angular Momentum Again? • And re-orienting: Rotate x y z L=r×p What was Angular Momentum Again? • And re-orienting: z L=r×p y x • Now we’re in a viewpoint that will be convenient to analyse Angular Momentum Operator • L is important to us because electrons are constantly changing direction (turning) when they are confined to atoms and molecules • L is a vector operator in quantum mechanics • Lx : operator for projection of L on a x-axis • Ly : operator for projection of L on a y-axis • Lz : operator for projection of L on a z-axis Angular Momentum Operator • Just for concreteness L is written in terms of position and momentum operators as: with Angular Momentum Operators • Ideally we’d like to know L BUT… One day this will be a lab… • Lx , Ly and Lz don’t commute! • By Heisenberg, we can’t measure them simultaneously, so we can’t know exactly where and what L is! Angular Momentum Operators • , • • and does commute with each of individually is the length of L squared. has the simplest mathematical form • So let’s pick the z-axis as our “reference” axis Angular Momentum Operators • So we’ve decided that we will use substitute for and as a • Because we can simultaneously measure: • L2 the length of L squared • Lz the projection of L on the z-axis BUT we can’t know Lx, Ly and Lz simultaneously! z L We’ve chosen to know only Lz (and L2) y Lz Ly Lx x Angular Momentum Operators • So we’ve decided that we will use substitute for and as a • Because we can simultaneously measure: • L2 the length of L squared • Lz the projection of L on the z-axis z L For different L2’s we’ll have different Lz’s y Lz can be anywhere in a cone for a given Lz x So what are the possible and eigenvalues and what are their eigenfunctions? Angular Momentum Eigen-System • Operators that commute have the same eigenfunctions • and commute so they have the same eigenfunctions • Using the commutation relations on the previous slides along with: we’d find…. One day this will be a lab too… Angular Momentum Eigen-System • Eigenfunctions Y, called: Spherical Harmonics • l = {0,1,2,3,….} angular momentum quantum number • ml = {-l, …, 0, …, l} magnetic quantum number Angular Momentum Vector Diagrams z Say l = 2 then m ={-2, -1, 0, 1, 2} For m =2 For m =2 Angular Momentum Vector Diagrams z • Take home messages: • The magnitude (length) of angular momentum is quantized: • Angular momentum can only point in certain directions: • Dictated by l and m Angular Momentum Eigenfunctions • The explicit form of and spherical polar coordinates: is best expressed in z q x r f For now, our particle is on a sphere and r is constant y • We won’t actually formulate these operators (they are too messy!), but their wave functions Y, will be in terms of q and f instead of x, y and z: • Yl,m(q,f) = Ql,m (q) Fm (f) Angular Momentum Eigenfunctions • l = 0, ml = 0 Angular Momentum Eigenfunctions l = 1 ml ={-1, 0, 1} Look Familiar? Angular Momentum Eigenfunctions l = 2 ml ={-2, -1, 0, 1, 2} Look Familiar? Particle on a Sphere r q can vary form 0 to p f can vary form 0 to 2 p r is constant Particle on a Sphere • The Schrodinger equation: 0 Particle on a Sphere • So for particle on a sphere: Legendre Polynomials Spherical harmonics • Energies are 2l + 1 fold degenerate since: • For each l, there are {ml} = 2l + 1 eigenfunctions of the same energy