Quantum Mechanics

Tutorials

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Planck Constant

Planck introduced a constant <math>h = 6.626 \times 10^{-34} J\ s\,</math>. It often shows up as <math>h/2\pi = \hbar = 1.055 \times 10^{-34} J\ s\,</math>.

Units

At the atomic scale, we generally put things in terms of constants such as:

• <math>m\,</math>, the mass of the particle in question.
• <math>c = 3.00 \times 10 ^{8} m/s\,</math>, the speed of light in a vacuum.
• <math>\hbar = 1.055 \times 10^{-34} J\ s\,</math>, the Planck constant.

The basic units are (for electrons with <math>m_e = 9.11 \times 10^{-28} g\,</math>):

• <math>mc^{2} = 0.51 MeV\,</math>, energy.
• <math>mc = 2.73 \times 10^{-25} g m/s\,</math>, momentum.
• <math>\hbar / mc = 3.9 \times 10{-14} m\,</math>, length.
• <math>\hbar / mc^{2} = 1.3 \times 10{-21} s\,</math>, time.

The Basic Idea

The basic idea of QM is:

• Light and matter behave both as particles of quantized energy and waves at the same time.
  • Matter, such as electrons, exhibit diffraction patterns, an artifact of its wave nature.
  • Light exhibit packets of quantized energy, such as in black body radiation, photoelectric effect, etc...
• The wave equation (below) contains all the information possible of how a particle will behave in the quantum world.
• If a particle doesn't change its wave equation, it doesn't change at all. That's why electrons spinning at the lowest energy level in an atom do not emit radiation.
• Particles can exist, simultaneously, in several different states. This is called quantum superposition. A silly way to understand what's really going on is to think of Schrodinger's cat. Since we do not know whether the atom has decayed and deployed the lethal gas in the box, we do not know whether the cat is alive or dead. All we can say is the cat is both alive and dead at the same time.
• The act of measurement causes the wave function to collapse, and the particle to choose a particle state depending on the type of measurement made.
• Because of the wave nature of the particle, we can only get so much detail about the position and momentum of a particle at any given moment. (Heisenberg Uncertainty Principle)

The Wave Equation

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What is a measurement?

Heisenberg postulated that measurements should really be called events, and we should think of them as being recorded in indelible ink in the history of the universe, irreversible once written. I like this explanation best. Everything is uncertain until we put pen to paper (or whatever device we've invented that will leave a record). Once this is done, uncertainty is lost.

How Particles Interact

Since we can't keep track of which particle is which, we must treat two particles as exactly identical. This has some implications about how particles interact. Fermions do not like to occupy the same state, while bosons prefer to.

When a particle's position or momentum is uncertain, and it creates an effect on other particles, that effect is also uncertain. Like the diffraction experiments, we must consider what happens as if the particle were everywhere and moving at all momentums, and multiply by the probabilities of each.

More to come!

I'm going to add more details as I remember them and as I re-read the source material and work through the examples and problems.

Source

I'm keeping detailed notes, as well as problem solutions, below.

• Quantum Physics by Stephen Gasiorowicz
• Introduction to Quantum Mechanics by David J. Griffiths

NOTE ON PROBLEM SOLUTIONS: If you want to be a physicist, you have to learn how to struggle with the math. Don't look at my solutions, or even my hints, until you have a solution of your own that you think is as good as it can ever be. Then let's compare notes and find the ideal solution together.