Course: PHY 305

Instructor: David Huse

F 2018

### Description of Course Goals and Curriculum

This course is a continuation of PHY 208. Its main goal is to introduce and develop techniques to solve for much more complex quantum systems compared to the ones discussed in PHY 208. In PHY 208 the student was asked to find the wave function of one particle for different scenarios corresponding to different potential functions (finite box, delta potential, a potential well, etc...). In PHY 305, we analyze the same problems but for systems of 2 particles and for time-dependent or complicated potentials. The 2-particle system is a great example to introduce concepts of quantum mechanics that give a deep insight into the probabilistic nature of the theory. In particular, they introduce the density matrix to differentiate between pure product states, entangled states and mixed states. The density matrix is an n*m matrix where the (n,m) element describes the probability to be in the (n,m) state. This provides a completely different way to approach quantum mechanics where the wave function can be totally forgotten and replaced by the density matrix. A fter this, the class moves on into studying different approximation methods to solve for systems with complicated or time-dependent potentials. The first method discussed is the variational method which provides a way to find the best upper bound for the ground state energy starting from a set of possible wave functions containing as much of the correct information as possible. The second method discussed is the WKB approximation. The method meant to solve for the wavefunction in regions where the Energy is different from the Potential which varies very slowly in comparison to the wavelength and therefore can be treated as constant. In regard to time-dependent potentials, this course introduced time-dependent perturbation theory and the adiabatic approximation. The first is very useful to understand the concepts that rely upon the transition between excited states of a system and find its foundations in time-independent perturbation theory already introduced in PHY 208. The second serves to solve problems in which the derivative of the potential is very small compared to V(t) but we it still results in large in V over time. This method is very useful to understand the differentiation between dynamical and geometric phases and realize how in quantum mechanics we cannot arbitrarily add a phase to the wavefunction. Indeed the integral around a loop of the geometric phase results in the so-called Berry's phase which is independent of the phase choice. Once we have fully exhausted all of this analysis, we use all these methods just described studying the significance and the conditions required for bound and scattering states to occur. The last week of the class is completely dedicated to Scattering theory where we discuss the 2 fundamental methods to solve for the scattering amplitude and consequently for its cross-section: First Born Approximation (used for weak potentials) and Partial wave analysis (spherically symmetric potentials).### Learning From Classroom Instruction

Huse's lectures strictly follow its lecture notes which he consistently posts on the blackboard at the end of every week. If on one side, one could think that this means that lectures are not that useful, I found them extremely important. Indeed some of the concepts introduced are very hard to grasp, in particular at first glance. This resulted in a lot of students asking many questions during the lecture and the answers given by the professor always provided with interesting insight into the material just discussed. The book is complementary to the lectures and I found extremely useful to skim through it before every class in order to be able to follow the professor and ask questions which could actually clarify my doubts. The best way to make sure you are keeping up with the pace of the class is to sit down and go over the lectures notes posted online, your notes and the textbook every week. I also found extremely useful to discuss the concepts introduced in class with other students in the course as this would not only help me clarify my own doubts but also help me to think about what was just discussed from a different perspective and therefore get more out of it.### Learning For and From Assignments

The problem sets are due weekly and can take from 5 to 15 hours. They can be extremely difficult but the whole class is meant to be that way. Students are expected to do a lot of thinking outside of class and therefore lectures are meant to not provide any single thing. Problem sets are not meant such that the student only needs to do the required work to solve them, in this class the professor actually expects more and the student who isn’t ready, willing and able to do more should be in a different course. This implies going to office hours, attending the problem sessions and generally discuss a lot with other peers in the course. It is crucial to understand fully how to solve each problem as the exams will strictly relate to them. The best way I found to tackle the problem sets was to go over my notes, lecture notes and textbook and then read the problems, really paying attention to why the professor would ask me to solve such a problem and what he wants me to get out of it. Then after spending a little time thinking about the general concepts behind each of the problems on my own, I would meet up with friends in the class and discuss the strategy to solve them and actually go over the actual, sometimes "nasty", math we would encounter. Office hours were a crucial time to get some tips and tricks to use to solve the problems but most importantly to hear the questions other students had and internalize the different approaches each student had on these problems. In regards to the exam, I think Huse is a very fair professor as his tests are strictly based on what was discussed in class and in the problem sets. Thus make sure you understand every problem you were assigned and with this, I don't mean that you should memorize them but you should be able to forget them, sit down and still be able to do them from scratch. Personally, to make sense of what was done during the semester I always make a 1-page recap for each of the topics discussed with tips on how to approach the problems and keywords to recall the most important concepts of that particular topic.### External Resources

The external resources for this class are very limited as quantum mechanics is a subject that can be studied in many different ways and therefore really depends on the approach your professor decides to implement. Online you can obviously find a lot of lectures from other institutes (for instance Stanford and MIT) but they won't be extremely helpful as Huse's approach is very different from what you will find in any website. That said, office hours together with the problem sessions will provide you with all the details and insights needed to succeed in this class.### What Students Should Know About This Course For Purposes Of Course Selection

This course is not a required course for physics majors. That said, I believe that students who are interested in pursuing a path in graduate school should strongly consider taking it as it represents a great introduction to advanced concepts of quantum theory which are a prerequisite to take graduate courses like quantum computing and quantum field theory. Also generally this course provides a much more abstract discussion of QM compared to PHY 208 and this kind of abstraction is where the physics education is going, therefore if you believe physics will constitute a central part of your life, take this course! it will be worth it! That said, remember that this is a hard course that requires a lot of time studying and thinking. The expectation is that the students will be eager to push forward in this and thus they won't be treated as in some other required course. Be aware of this, understand your motive to take this class and be ready to work hard and learn a ton!Introduction to the Quantum Theory