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Quantum Mechanics: Spherical Harmonics and Schrödinger Equation

Learn about spherical harmonics and the Schrödinger equation in quantum mechanics, including the separation of variables and the normalization of wave functions.

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Quantum Mechanics: Spherical Harmonics and Schrödinger Equation

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  1. Physics 451 Quantum mechanics I Fall 2012 Oct 31, 2012 Karine Chesnel

  2. Phys 451 Announcements • Homework this week: • HW #16 Friday Nov 2 by 7pm

  3. Pb 4.1 Phys 451 Position- momentum in 3 dimensions

  4. z Each stationary state verifies y x Phys 451 Schrödinger equation in 3 dimensions

  5. Infinite cubical well: V=0 inside a box Pb 4.2 Separation of variables Phys 451 Schroedinger equation in cartesian coordinates

  6. z r y x The angular equation The radial equation Phys 451 Schrödinger equation in spherical coordinates

  7. z r Further separation of variables: y x m integer (revolution) F equation: q equation: Phys 451 The angular equation

  8. z r y x Solution: Legendre function Legendre polynomial Physical condition l,m integers Phys 451 The angular equation

  9. z r Legendre function Solution: y x are polynoms in cosq (multiplied by sinq if m is odd) Phys 451 The angular equation

  10. l Azimuthal quantum number m Magnetic quantum number Phys 451 Spherical harmonics z r y x Simulation: www.falstad.com

  11. z r y x 1. Legendre polynomial 2. Legendre function Pb 4.3 3. Plug in cosq Quantum mechanics Spherical harmonics Method to build your spherical harmonics: 4. Normalization factor

  12. Phys 451 Quiz 21 Which one of the following quantities could not physically correspond to a spherical harmonic? A. B. C. D. E.

  13. Change of variables The radial equation Centrifugal term Phys 451 The radial equation z r y x Form identical to Schrödinger equation ! with an effective potential

  14. Radial part Angular part or Phys 451 Normalization z r y x

  15. Phys 451 Orthonormality z r y x Spherical harmonics are orthogonal Angular part Pb 4.3 and 4.5

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