Chap 12 Quantum Theory: techniques and applications Objectives: Solve the Schrödinger equation for: Translational motion (Particle in a box) Vibrational motion (Harmonic and anharmonic oscillator Rotational motion (Particle on a ring & on a sphere

Rotation in three-dimensions: a particle on a sphere Hamiltonian: Schrodinger equation Laplacian V = 0 for the particle and r is constant, so By separation of variables:

Fig 9.35 Spherical polar coordinates. For particle on the surface, only θ and φ change.

The Schrödinger equation for a particle on a sphere where the laplacian (operator): and the legendrian (operator): Through separation of variables: Ψ(θ,φ) = Θ(θ)Φ(φ) The solutions then are: The spherical harmonics In spherical polar coordinates:

Table 9.3 The spherical harmonics Y l,ml (θ,φ)

Fig 9.34 Wavefunction for particle on a sphere must satisfy two boundary conditions Therefore: two quantum numbers l and m l where: l ≡ orbital angular momentum QN = 0, 1, 2,… and m l ≡ magnetic QN = l, l -1,…, - l

Fig 9.38 Space quantization of angular momentum for l = 2 Problem: we know θ, so... we can’t know φ θ Because m l = - l,...+ l, the orientation of a rotating body is quantized! Permitted values of m l

Fig 9.40 Space quantization of angular momentum for l = 2 for which φ is always indeterminate. θ Angular momentum can take 2( l +1) orientations The vector model of angular momentum:

Fig 9.39 The Stern-Gerlach experiment confirmed space quantization (1921) Ag Classical expected Observed Inhomogeneous B field Classically: A rotating charged body has a magnetic moment that can take any orientation Quantum mechanically: Ag atoms have only two spin orientations Configuration of Ag atoms: [Kr] 4d 10 5s 1

Fig 9.41 Space quantization of electron spin for which φ is always indeterminate. θ = 55° Quantum mechanically: Ag atoms have only two spin orientations. But... angular momentum But... angular momentum can take 2( l +1) orientations. 2( l +1) = 2 only if l = ½, 2( l +1) = 2 only if l = ½, contrary to the conclusion that l must be an integer! Angular momentum must arise Angular momentum must arise from spin, not orbital motion.

Angular momentum in the Stern-Gerlach experiment must arise from spin, not orbital motion To distinguish spin angular momentum from orbital angular momentum, we use: To distinguish spin angular momentum from orbital angular momentum, we use: Spin quantum number, m s Spin quantum number, m s where m s = s, s-1,... –s Electrons, protons, & neutrons have s =½ Electrons, protons, & neutrons have s =½ H-1 has s = ½, but H-2 has s = 1 (NMR) H-1 has s = ½, but H-2 has s = 1 (NMR) C-12 has s = 0, but C-13 has s =½ (NMR) C-12 has s = 0, but C-13 has s =½ (NMR) Photons have s = 1 Photons have s = 1 Fermions → particles with half-integral spins Fermions → particles with half-integral spins Bosons → particles with integral spins Bosons → particles with integral spins

Exam #1 Friday 2 Oct 2009 Part 1: Closed book, closed notes Pick up: In envelope to left of my door Due: By 1600 Friday Part 2: Open book, open notes Due: By 1600 Wednesday 7 Oct 2009