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

Rotational Motion in 2-D Fig 9.27 Angular momentum of a particle of mass m on a circular path of radius r in xy-plane. Classically, angular momentum: J z = ±mvr = ±pr and Where’s the quantization?!

Fig 9.28 Two solutions of the Schrödinger equation for a particle on a ring For an arbitrary λ, Φ is unacceptable: not single-valued: Φ = 0 and 2π are identical Also destructive interference of Φ This Φ is acceptable: single-valued and reproduces itself.

Acceptable wavefunction with allowed wavelengths:

Apply de Broglie relationship: Now: J z = ±mvr = ±pr As we’ve seen: Gives: where m l = 0, ±1, ±2,... Finally: Magnetic quantum number!

Fig 9.29 Magnitude of angular moment for a particle on a ring. Right-hand Rule

Fig 9.30 Cylindrical coordinates z, r, and φ. For a particle on a ring, only r and φ change Let’s solve the Schrödinger equation!

Fig 9.31 Real parts of the wavefunction for a particle on a ring, only r and φ change. As λ decreases, |m l | increases in chunks of h

Fig 9.32 The basic ideas of the vector representation of angular momentum: Vector orientation Angular momentum and angle are complimentary (Can’t be determined simultaneously)

Fig 9.33 Probability density for a particle in a definite state of angular momentum. Probability = Ψ*Ψ with Gives: Location is completely indefinite!

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.

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

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

Fig 9.36 Wavefunctions for particle on a sphere Sign of Ψ

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 9.39 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.

Fig 9.40 Space quantization of angular momentum for l = 2 where φ is indeterminate. θ