You all have a computer (or a phone) loaded with ``Applications." You should therefore have no problem defining ``Application." Its a thing, right? It does something? Does an application on a phone mean the same thing as an application of quantum mechanics? What would you like to see in a course on ``Applications of Quantum Mechanics?" For some people an ``application of physics" is something fairly narrowly defined (for example, something you can sell like a computer chip). Here we are a bit broader. For us, ``applications" mean that we are using quantum mechanics to describe the physical world, as opposed to studying the mathematical structure. (Projectile motion is an ``application" of Newton's laws.) Along the way, you will undoubtedly get a significantly deeper understanding of the underlying formalism.
Review: Quantum Mechanics
In this section, I will review some of the main pieces of quantum mechanics that you know. I want to caution you that this section is backwards compared to everything else in the course, since it starts with the rules. In the future I hope to start with physical settings, then figure out the rules. This latter approach reflects the way Science really works: an ongoing effort to develop and refine a series of working definitions. The axiomatic approach, however, is nice for summarizing things after-the-fact. Activity: Complete Single Particle Quantum Mechanics handout
Rules of Single Particle Quantum Mechanics in 1D
The object of interest in single particle quantum mechanics is the wavefunction $\psi(x,t)$. It encodes information about the position of a particle at time $t$. The rules by which this information is encoded are:
- If you measure the position of the particle, you will find it somewhere
- If you measure it twice in quick succession, you will find it in the same place each time (though if you wait too long, it will probably fly away -- the measurement process often gives the particle a kick.
- The probability of finding it at position $x$ is $P(x,t)=|\psi(x,t)|^2$. More precisely, the probability of finding a particle within $dx$ of position $x$ is $P(x,t) dx = |\psi(x,t)|^2 dx $
If the atom has mass $m$, and is in a potential well $V(x)$, the wavefunction evolves via Schrodinger's equation $$i\hbar\partial_t \psi(x,t)=-\frac{\hbar^2 \partial_x^2}{2m} \psi(x,t)+V(x) \psi(x,t). $$ One of our goals in this course is to really understand what this means -- particularly incases of physical interest. IE. We are not concerned with the abstract properties of solutions of this equation, rather we care about the real properties for physical $V$. In order to we will use some of the following tools to understand this equation:
- Dimensional Analysis and Scaling
- Variational techniques
- Symmetry Analysis (if you are pretentious you can instead say ``group theory" but that makes it sound too formal -- I promise to never write down the axioms of group theory)
- Computer simulations
Observables
Most measurements are associated with a Hermitian operator $\hat O$. [In fact traditionally all measurements were associated with a Hermitian operator.] Some concrete examples are position, momentum, and energy. These are associated with Hermitian operators $\hat x,\hat p,\hat H$, defined by: $$ (\hat x \psi)(x)=x \psi(x)$$ $$ (\hat p \psi)(x)=-i\hbar \psi^\prime(x)$$ $$ (\hat H \psi)(x)= \frac{-\hbar^2}{2m} \psi^{\prime\prime}(x)+V(x)\psi(x).$$ The operators $\hat x$, $\hat p$ and $\hat H$ are all linear operators. Teaser: In a very real sense, these operators can be thought of as matrices. In your second recitation section you will see one approach to explicitly calculating these matrices. Whenever things start seeming abstract, you can always return to the matrix representation of the operators. Activity: Complete Linear Operator handout In any measurement of $\hat O$ on a single isolated quantum system, one will find a value $ o_n$ with probability $P_n$. The allowed values of $ o_n$ are given by the eigenvalues of $\hat O$. Lets assume that the eigenvalues of $\hat O$ are non-degenerate. The eigenvectors are then uniquely defined by $$ \hat O \phi_n = o_n \phi_n.$$ The probability of finding a given value of $O$ is $$ P_n =\left| \int \!\!dx\,\phi_n^*(x) \psi(x)\right|^2.$$ Often we do ensembles of measurements (either we have an ensemble of systems we are measuring, or we repeat a measurement over and over). Then it is useful to as what is the average (mean) value of the observable. We call this the "expectation value" and denote it $ \langle \hat O \rangle$. By definition $$\langle \hat O \rangle=\sum_n o_n P_n= \sum_n o_n \left|\int \!\! dx \phi_n(x) \psi(x)\right|^2. $$This can be simplified to $$ \langle \hat O\rangle=\int \!\!dx\,\psi^*(x) {\hat O}\psi(x).$$ Note, this is the mean of a number of measurements. It may be that no individual measurement will yield that value. [Imagine that a particle is at position $ x=-1$ with probability $ 1/2$ and $ x=1$ with probability $1/2$. Clearly $\langle x \rangle =0$, but one will never find the particle at $x=0$.] From a classical perspective there is nothing intuitive about how measurement works in quantum mechanics. It is just strange. Later we will more precisely model the measurement process. At some point, however, there will always be some magic where we rely on this axiomatic framework. [As a good trick, often we can skip all the modeling and just apply the axiom.] From a generic standpoint, spectroscopy measures the energy of the system. More precisely spectroscopy measures the energy that a probe is able to add (or remove) from a system. We are not quite ready to write down a Hermitian operator which gives the optical absorption spectrum of an atom (but we will). At this point, however, I just want you to appreciate that one can interpret the discreteness of atomic absorption spectra as a consequence of the way measurement works in quantum mechanics.
One final bit. Typically after a measurement of operator $\hat O$ which yields value $o_n$, the wavefunction becomes $\phi_n$. Physically this means that two very closely spaced measurements will yield the same value.
Class Structure
Although I have taught this course a couple times , many of my decisions about class structure or content still evolving. I want to be flexible: it is your job to tell me what is working/not working so I can adjust.
Lectures:
Tuesdays and Thursdays -- introduce material and practice techniques. There will be in-class activities. These should be handed in with the homework.
Recitations:
Fridays -- In computer room (but not always doing computer exercises). Fairly structured, but will include some time to ask questions about homework or other class material. We will do some computer exercises using IPython Notebooks. The activity sheets should be handed in at the next recitation.
Web Site:
I am using a custom built web site for all course matterial, but will use Blackboard for handing in electronic documents.
Homework:
We will vote on day which homework will be due (one of the class days -- Tuesday, Thursday, or Friday). Last year Fridays won, so by default we will use that. Office hours will be held the day before, and we will schedule a "Homework Party" time/location. [Last year we turned most of Friday afternoon into such a party.] Feel free to work together, but independently write your solutions.
Final Exam
We will give an open book take home exam. Unless there are any objections, this will be handed out on Tuesday Nov 26. It will be due at our official exam period.