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Physics 451. Quantum mechanics I Fall 2012. Oct 29, 2012 Karine Chesnel. Phys 451. Announcements. Homework this week: HW #16 Friday Nov 4 Pb 4.1, 4.2, 4.3, 4.5. z. y. x. Phys 451. Schrödinger equation in 3 dimensions. Pb 4.1. Phys 451. Quiz 18.
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Physics 451 Quantum mechanics I Fall 2012 Oct 29, 2012 Karine Chesnel
Phys 451 Announcements • Homework this week: • HW #16 Friday Nov 4 • Pb 4.1, 4.2, 4.3, 4.5
z y x Phys 451 Schrödinger equation in 3 dimensions
Pb 4.1 Phys 451 Quiz 18 Which one of the following quantities is not correct? A. B. C. D. E.
Pb 4.1 Phys 451 Position- momentum in 3 dimensions
now Laplacian Phys 451 Schrödinger equation in 3 dimensions
General solution Each stationary state verifies Phys 451 Schrödinger equation in 3 dimensions
radius polar angle azimuthal angle Laplacian Phys 451 Spherical coordinates z r y x
z r y x Separation of variables Phys 451 Schrödinger equation in spherical coordinates
z r y x The angular equation The radial equation Phys 451 Schrödinger equation in spherical coordinates
z r Further separation of variables: y x m integer Azimuthal equation: q equation: Phys 451 The angular equation
z r y x Solution: Legendre function Legendre polynomial Physical condition l,m integers Phys 451 The angular equation
z r Legendre function Solution: y x are polynoms in cosq (multiplied by sinq if m is odd) Phys 451 The angular equation
l Azimuthal quantum number m Magnetic quantum number Phys 451 Spherical harmonics z r y x Normalization Simulation: www.falstad.com/qmatom/