Quantum Mechanics in three dimensions

Zeeman Splitting

The Bohr Atom Classical electromagnetism does not hold for atom sized systems. Used Planck’s energy quantization ideas to postulate that electrons orbit in fixed, stable, nonradiating states, given by Used Einstein’s concept of a photon to define the frequency of radiation emitted when an electron jumps from one state to another. The photon energy is just the energy difference between states, i.e., Used classical mechanics to calculate the orbit of the electron.

The Schrodinger Equation The time dependent Schrodinger equation: can be “separated” to get the time-independent Schrodinger equation which can be used to find the “stationary states” or standing waves in a potential.

The time-independent Schrodinger equation in 3 dimensions “Laplacian” Can we use our previous knowledge to guess some of the characteristics of a particle in a 3 dimensional “box”? What are the boundary conditions? What is the form of the wave function? Can you deduce anything about the ground state? Higher states?

The Schrodinger Equation in Three Dimensions “Laplacian”

Particle in a 3-dimensional box U=0 inside the box Leads to “degenerate” states: unique states with the same energy! Also for 3D Harmonic oscillator

A visualization: two dimensional box First Excited State Ground State Second Excited State

Spherical coordinates …make the most sense when describing atoms. f r q

The Schrodinger Equation in Spherical Coordinates conversion from cartesian coordinates to spherical polar coordinates Laplacian in spherical polar coordinates: The Schrodinger equation in spherical polar coordinates:

The polar solution The polar part of the Schrodinger equation is: With some rearrangement, this can be recognized as the associated Legendre equation: Luckily, someone has already solved this equation, so we don’t have to:

The spherical harmonics

So what do these functions look like So what do these functions look like? Let us look at them by a polar plot – then let us look at the magnitude

But these functions live in 3 dimensions – simple add phi – do these look familiar?

Have you seen these shapes before In say….chemistry?

Those weird – s,p,d,f orbitals that made chemistry easy to understand came from the…… That horrible messy 3D Schrodinger equation!!

Quantization of Angular momentum +1 +2 +3 +4 m -1 -2 -3 -4 Space quantization

The Bohr Atom Revisited Classically: Bohr figured out that angular momentum was actually quantized: The Schrodinger equation in three dimensions gives us another insight as to why that is: So the Bohr model was kindda right – but not really completely for example he was rught that there is angular momentm and it is quantized But now some states have 0 angular momentum!

The Solar Magnetic Field The atomic spectra tells us a bit more about the solar spectrum then from the simple Bohr picture. For example if we look closely at the sun closely we can see it that it not just a uniform disc but it has solar spots with magnetic field in jets. Also – as we talked about before, the atomic spectra shows up as absorption (dark) lines in the solar spectrum. As one zooms into this absorption spectrum near a solar spot we see that the line splits into three lines. This can be understood in terms of the angular momentum of an L=1 orbital. When an L=1 orbital is subject to a magnetic field, it’s energy splits into three levels m_l=-1,0,+1. This is called Zeeman splitting. Spectral lines split into three Because of (normal) Zeeman effect