1 / 14

Exploring Schrödinger Equation in 3D: Quantum Mechanics Concepts in Spherical Coordinates

This lecture covers the Schrödinger equation in 3 dimensions, focusing on the general solutions, radial and angular equations, and concepts like azimuthal angle and separation of variables. Students will delve into the mathematical aspects of quantum mechanics, specifically in the context of spherical coordinates. The session includes a discussion on Legendre functions, polynomial solutions, normalization, and introduces the azimuthal and magnetic quantum numbers. Further exploration involves simulations to deepen understanding and practical application.

bmatteson
Download Presentation

Exploring Schrödinger Equation in 3D: Quantum Mechanics Concepts in Spherical Coordinates

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Physics 451 Quantum mechanics I Fall 2012 Oct 29, 2012 Karine Chesnel

  2. Phys 451 Announcements • Homework this week: • HW #16 Friday Nov 4 • Pb 4.1, 4.2, 4.3, 4.5

  3. z y x Phys 451 Schrödinger equation in 3 dimensions

  4. Pb 4.1 Phys 451 Quiz 18 Which one of the following quantities is not correct? A. B. C. D. E.

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

  6. now Laplacian Phys 451 Schrödinger equation in 3 dimensions

  7. General solution Each stationary state verifies Phys 451 Schrödinger equation in 3 dimensions

  8. radius polar angle azimuthal angle Laplacian Phys 451 Spherical coordinates z r y x

  9. z r y x Separation of variables Phys 451 Schrödinger equation in spherical coordinates

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

  11. z r Further separation of variables: y x m integer Azimuthal equation: q equation: Phys 451 The angular equation

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

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

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

More Related