Quantum Mechanics of the Hydrogen Atom
This video gives an overview of some of the different quantum mechanical descriptions of the hydrogen atom that have been developed over the years, especially the Bohr atom, and solving the Schrödinger equation to get the wavefunctions and resulting probability density functions for the electron in a hydrogen atom.
This video is part of my Quantum Mechanics for Physical Chemistry playlist: • Quantum Mechanics
Timestamps:
00:00 start
00:08 hello and suggestions for how best to learn this
02:43 lecture begins
03:53 Balmer-Rydberg-Ritz equation and the H emission spectrum
08:21 The Bohr atom: comparison to the orbit of the moon
08:25 The Bohr atom: derivation from assuming momentum is quantized
18:13 Problems with the Bohr atom
21:27 Defining the Hamiltonian for the Schrödinger equation of the H atom
28:18 The kinetic energy term in the Hamiltonian for the H atom
32:06 Spherical coordinates
35:39 Deriving the Laplacian in Cartesian coordinates
40:47 The Laplacian in spherical coordinates
51:38 Incorporating the Laplacian into the Schrödinger equation of the H atom
52:35 Defining the complete Schrödinger equation of the H atom
53:21 Solving the Schrödinger equation of the H atom
54:10 Separability applied to solving the Schrödinger equation of the H atom
1:02:00 Solving the angular portion of the Schrödinger equation of the H atom
1:06:46 Solving the radial portion of the Schrödinger equation of the H atom
1:16:18 Combining the angular and radial solutions to get complete wavefunctions
1:20:01 Constraints on the quantum numbers n, l, and m_ℓ and degeneracy
1:24:44 What about the spin quantum number (m_s)?
1:30:54 Summary: the Bohr atom
1:32:53 Summary: the potential energy term of the Hamiltonian
1:34:32 Summary: the kinetic energy term of the Hamiltonian
1:35:35 Summary: writing the Schrödinger equation of the H atom
1:37:07 Summary: solving the Schrödinger equation of the H atom
1:44:28 Summary: constraints on the quantum numbers
1:47:07 Summary: gaining some perspective
If you have never solved the time-dependent Schrodinger equation for the 1-dimensional particle, I definitely recommend that you watch this video first:
• Solving the Time-dependent Schrödinger Equ...
Here is a link to the video by Andrew Meyertholen that I mentioned gives a nice derivation of the Laplacian operator in spherical coordinates:
• Gradient and Laplacian in Spherical Coordi...