Quantum mechanics I Fall 2012 Physics 451 Quantum mechanics I Fall 2012 Nov 9, 2012 Karine Chesnel

HW #18 today Nov 9 by 7pm Homework next week: Phys 451 Announcements HW #18 today Nov 9 by 7pm Homework next week: HW #19 Tuesday Nov 13 by 7pm HW #20 Thursday Nov 15 by 7pm

The hydrogen atom Phys 451 How to find the stationary states? Step1: determine the principal quantum number n Step 2: set the azimuthal quantum number l (0, 1, …n-1) Step 3: Calculate the coefficients cj in terms of c0 (from the recursion formula, at a given l and n) Step 4: Build the radial function Rnl(r) and normalize it (value of c0) Step 5: Multiply by the spherical harmonics (tables) and obtain 2l +1 functions Ynlm for given (n,l) (Step 6): Eventually, include the time factor:

Phys 451 The hydrogen atom Representation of Bohr radius

The hydrogen atom Expectation values Pb 4.13 Most probable values Quantum mechanics The hydrogen atom Expectation values Pb 4.13 Most probable values Pb 4.14

Expectation values for potential Quantum mechanics The hydrogen atom Expectation values for potential Pb 4.15

Phys 451 The angular momentum Pb 4.19

Phys 451 The hydrogen atom Anisotropy along Z axis Representation of

The angular momentum Phys 451 Ladder operator If eigenvector of L2, then eigenvector of L2, same eigenvalue If eigenvector of Lz with eigen value m then eigenvector of Lz, new eigenvalue

The angular momentum Pb 4.18 Phys 451 Ladder operator Eigenstates Top Value =+l Bottom Value = -l Ladder operator Eigenstates Pb 4.18

Quiz 25 Phys 451 When measuring the vertical component of the angular momentum (Lz ) of the state , what will we get? A. 0 B. D. E.

in spherical coordinates Phys 451 The angular momentum in spherical coordinates x y z r

In spherical coordinates Phys 451 The angular momentum In spherical coordinates x y z r Pb 4.21, 4.22

The angular momentum eigenvectors Phys 451 y z r x and were the two angular equations for the spherical harmonics! Spherical harmonics are the eigenfunctions

and Schrödinger equation Phys 451 The angular momentum and Schrödinger equation x y z r 3 quantum numbers (n,l,m) Principal quantum number n: integer Azimutal and magnetic quantum numbers (l,m) can also be half-integers.