## Wednesday, July 8, 2009

### Quantum Mechanics

Quantum Mechanics is one of the most popular yet misunderstood physics topics out there. There are many myths around quantum mechanics that I run into from time to time, and I thought I'd devote some posts to the topic.

Perhaps the biggest myth surrounding quantum mechanics is the idea that it doesn't make sense. This idea is absurd. Quantum mechanics describes how our world works- if it doesn't make sense, then you just don't understand it. Or at least you haven't thought about it in the right way.

Quantum mechanics is baffling yet incredibly simple. You can literally write down all of quantum mechanics on a half-sheet of paper. As a matter of fact, here it is:

- The state of a system is entirely represented by its wave function, which is a unit vector of any number of dimensions (including infinite) existing in Hilbert space. The wave function can be calculated from the Schrodinger equation:
$i\hbar {\partial \psi (x, t) \over \partial t} = -\frac{\hbar^2}{2 m} \frac{\partial^2 \psi (x, t)}{\partial x^2} + U(x) \psi (x, t)$
- Observable quantities (like position, momentum) are represented by Hermitian Operators, which function as linear transformations that operate on the wave function.
- The expectation value (in a statistical sense) of an observable quantity is the inner product of the wave function with the wave function after being operated on by the observable's hermitian operator.
- Determinate States, or states of a system that correspond to a constant observed value, are eigenstates of the observable's hermitian operator, while the observed value is the eigenvalue. (ex. energy levels that give rise to discrete atomic spectra are eigenvalues corresponding to energy determinate states.)
- All determinate states are orthogonal and all possible states can be expressed as a linear combination of determinate states.
- When a measurement is made, the probability of getting a certain value is the square root of the inner product of that value's determinate state with the wave function.
- Upon measurement, the wave function "collapses", becoming the determinate state corresponding to the value that was measured.

Okay, so this is probably confusing to those of you who haven't had a class in quantum mechanics or an advanced course in linear algebra. Getting passed the math, it really isn't that hard conceptually. The important thing to note, though, is the fact that it can be defined so concisely. I think I've actually included more than necessary, so it probably can be even more concise than what I've written. Quantum mechanics is pretty complex in application, but is simple at its core. All of the best theories have this quality.

I'll probably post some stuff later that will (hopefully) clear up some of the details.

Another myth I run into is around the term "Quantum Physicist". There isn't such thing- at least in the professional sense. The reason why is the fact that there isn't a physicist in the world who doesn't use quantum mechanics. If the term "Quantum Physicist" represents a scientist who uses quantum mechanics in his/her research, we can probably just agree to just use the term "Physicist". Likewise, it is impossible to go to college and major in "Quantum Physics". Any respectable university would require its physics majors to learn quantum mechanics, so there is no reason to create a new major around it. I'm saying this in reference to the nerdy characters in movies and TV shows who are described using these terms. If you know anyone who writes screenplays, let them know.

I was going to get into some of the misunderstandings around actual quantum mechanics, but maybe I'll get into it later. Most of these misunderstandings involve the indeterminacy of the statistical interpretation, the Heisenberg uncertainty principle, and the collapse of the wave function. I'll try to get to all these issues later. For now, I'm hungry.