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

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

Phys 451 Position- momentum in 3 dimensions Pb 4.1

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

Schroedinger equation in cartesian coordinates Phys 451 Schroedinger equation in cartesian coordinates Pb 4.2 Infinite cubical well: V=0 inside a box 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 F equation: m integer (revolution) 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 Azimuthal quantum number Magnetic quantum number l m Simulation: www.falstad.com

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

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

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

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

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