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

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

Phys 451 Schrödinger equation in 3 dimensions x y z

Which one of the following quantities 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 Pb 4.1

Phys 451 Schrödinger equation in 3 dimensions now Laplacian

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

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

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

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

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

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

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

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