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Chap 12 Quantum Theory: techniques and applications

Chap 12 Quantum Theory: techniques and applications. Objectives: Solve the Schr ö dinger 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.

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Chap 12 Quantum Theory: techniques and applications

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  1. Chap 12Quantum 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)

  2. 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: Jz = ±mvr = ±pr and Where’s the quantization?!

  3. Fig 9.28 Two solutions of the Schrödinger equation for a particle on a ring • For an arbitraryλ, 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.

  4. Acceptable wavefunction with allowed wavelengths:

  5. Apply de Broglie relationship: Now: Jz=±mvr =±pr As we’ve seen: Gives: where ml = 0, ±1, ±2, ... Finally: Magnetic quantum number!

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

  7. 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!

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

  9. 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)

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

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

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

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

  14. Table 9.3 The spherical harmonics Yl,ml(θ,φ)

  15. Fig 9.36 Wavefunctions for particle on a sphere + Sign of Ψ + - - + -

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

  17. Fig 9.39 The Stern-Gerlach experiment confirmed space quantization (1921) Classical expected Observed Ag 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.

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

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