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Ideas on Quantum Theory Curriculum

  1. Jul 23, 2009 #1
    So I've been given the opportunity to create my own undergraduate honours program. I can take whatever courses from whatever faculty I want at almost any university (although I'm keeping it local). I can create from scratch courses that do not yet exist immediately prior to taking them. I am to write a thesis before graduation, and have access to researchers at the Institute for Quantum Computing and the Perimeter Institute for Theoretical Physics. The program is extremely research-intensive, both in studies and in "field work".

    My interests are in the philosophy of physical theory, mathematical physics, quantum information, and emergence in physical theories and models. In particular, I am interested in modeling quantum mechanical molecular systems and interactions using algebraic and geometric methods.

    I am looking to set up a preliminary curriculum to give the academic advisors an idea of where I'm looking to go before I step foot in the door.

    Of course, as I traverse the ocean, I must sway with the seas, but as yet even my preliminary course is not mapped. I want to emphasize lateral thinking (whatever that means). Currently, I am considering the following split:

    I - Information and Computation, e.g.:
    probability theory
    statistics
    information theory
    theory and models of computation

    II - Analysis and Algebra, e.g.:
    abstract algebra (e.g group theory)
    multilinear algebra
    complex analysis / Hilbert Spaces

    III - Geometry, e.g.:
    metric spaces
    forms/vector analysis/multivariable calculus
    topology
    manifolds

    IV - Physics, e.g.:
    mechanics
    statistical mechanics
    quantum theory
    quantum information and computation

    V - Computational Modeling and Approximation Theory

    I am thinking of doing my thesis on the application of geometric methods to quantum mechanical systems.

    There are likely many factors I haven't considered, and so I am wondering if the kind people of this forum could help me think about this!

    Thanks in advance!
     
  2. jcsd
  3. Jul 23, 2009 #2
    how in the hell do you get to do this? do you already know single variable calc and odes and linear alg and real analysis

    what are geometric methods in quantum mechanical systems?
     
  4. Jul 23, 2009 #3
    I'm still wondering how more people don't know about the idea of an independent study. Many universities offer it.

    I've taken four calculus courses, and a linalg course.

    I'm not entirely sure how easily I can elucidate that right now.
     
  5. Jul 23, 2009 #4

    Hey, a fellow Waterlooer (well, I was a waterlooer). I'm afraid I really don't know what you're talking about here. You're trying to construct a new undergrad program for what exactly? I mean to design an undergrad program you need to be specific about what you want someone in that program to do. There's also the structure of the courses/university to consider. If you're talking about Waterloo, the kind of content you're talking about would have you taking PMATH, AMATH, CS, PHYS and engineering courses all of which have pre-req's and co-req's and anti-req's. A little more clarification would be helpful. I know Waterloo's undergrad calendar well enough that I could probably help you with a specific course sequence I just don't really understand what you're trying to specifically learn/accomplish.
     
  6. Jul 23, 2009 #5
    Basically, I'm looking at switching into the Independent Studies program, which grants you the flexibility of taking almost any course you want (sans engineering courses). To do this, I must come up with, not a course list, but a "study plan", which is a more vague variant on a full-blown curriculum. It is this plan that I will present first to a committee of entrance advisors, and it is this plan that I will at least partially adhere to as I trek through.

    An interesting note is that I can completely bypass the requisites. This is a benefit of being in IS. The downside is that it would involve a *lot* of course override forms, and a lot of "convincing" of professors.

    As far as what I am trying to learn: I have a vivid mental image of it: it is best summarized as the study of emergent properties of quantum-mechanical systems. There would be applications to nanotechnology, quantum computing, molecular modeling, and the like. For me, mathematics is the study of properties emergent from "rules", and hence I require it not only as a tool for computation, but also to gain deep insight into this process.
     
  7. Jul 23, 2009 #6
    K, I'd like to preface this by saying that such a program sounds like it would be nearly impossible to explain to a graduate admissions commitee. So if you want to do grad school I'd rethink things (or, at the very least, make sure you make a LOT of fans amongst professors). However, what you're looking for is very similar to my own background. I did "Computational Science/Physics Specialization" with two unclaimed minors in CS and AMATH (Waterloo math has a no double dipping policy for claiming majors(i.e. you can't apply one course to more than one program/major/minor even it's a common requirement) so you might want to look at the curriculum for that for a guideline.

    For what you want to do I'd strip most/all of the pure math stuff out, algebraic theory and such. It's not that there's not applications of this but there is little benefit in going through an entire textbook written for pmathies rather then just learning it as it becomes relevant to physics. How would you like suggested studying outlined? By course or by textbook or what?
     
  8. Jul 23, 2009 #7
    This was my first and still is my primary concern. However, I intend on pursuing many research positions during the course of my studies, and will be writing a thesis at the end of it. I am running the idea by some members of the IQC next week to ensure that I am not digging my own grave.

    Understood and agreed with respect to the PMath, except for the applicability to emergence (perhaps). I should probably just make up a course on emergence phenomena and have units on mathematics and applications.

    I would love a textbook list to get me started. However, the details are not nearly as important as the philosophy or strategy of the curriculum.

    When did you graduate, out of curiosity?
     
  9. Jul 23, 2009 #8
    Only last yearish. I myself am really interest in emergent phenomena. I almost did my grad at the university of calgary's "Complexity Science" institute. However, scientists who specialize in "emergent phenomena" in general (as opposed to a specific field) have a reputation for getting trumped by those who specialize in a particular field (i.e. jack of all trades, master of none). If I were to do it all again and prescribe a path/degree I'd probably say something like (i'm going to mix textbooks and waterloo courses but you can always just look at the prescribed textbook for a given course):

    Classical Mechanics:
    -Goldstein

    Quantum Mechanics:
    -Griffiths
    -Le Bellac (I'm kinda torn for prescribing a middle range quantum book since they're all kinda mediocre)
    -Sakurai (this is the standard grad text in grad school)

    Statistical Mechanics:
    -... (stat mech is very important but I've never seen a GREAT undergrad book, but try Schroeder (Carter is crap))

    E&M:
    -Griffith (not the same book as QM)
    -Jackson (this is the graduate level book of choice)

    Math:
    -Calculus 1-4
    -Complex Analysis
    -Linear algebra 1 & 2
    -Mathematical Physics 1&2 (the textbook here is Boas)

    CS (i'm just going to list Waterloo courses here):
    -CS 123,124, 230, 370 (dont' bother with something like 476, it sucks, I took it)
    -PHYS 239 (if you've done CS 123,124 don't bother with PHYS 139) and 339.

    Then snag yourself an undergrad thesis project with a good prof (I can suggest waterloo profs in private messaging if you want) who is doing work you are interested in.
     
  10. Jul 23, 2009 #9

    jtbell

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    Staff: Mentor

    If you want to go to grad school, make sure you cover all the core areas of physics. I'd add a couple of semesters of E&M at the level of Griffiths. You might also consider optics, which would lead into quantum optics.
     
  11. Jul 23, 2009 #10
    Griffith's E&M actually covers optics. I always found this curious since NO ONE likes Hecht. Why not use Griffiths?
     
  12. Jul 24, 2009 #11

    Vanadium 50

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    Staff Emeritus
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    Education Advisor

    I would recommend that you figure out what it means before making your proposal. Otherwise, you might find yourself in a difficult position.
     
  13. Jul 24, 2009 #12
    If you want to do research in physics, do an undergraduate physics degree! I did a "hotch potch" degree like this, because I was interested in all sciences and thought I'd take a "combined science" degree that would let me chop and change. I quickly realised that no research group would take me seriously in *any* science if I had such a general degree and cobbled the degree together into "something like" a physics degree. But in doing that I ended up doing maths courses that had no connection to physics and "Astronomy for idiots" when I could have handled an advanced course. I missed out on the more interesting physics courses 'cause they were only available for single honours. Professors did not treat me seriously because I was a "combined student". Failing medical & physics students were dumped into "combined", completely lowering the respect given to the degree. Only 2 people out of 200 got first class honours in "combined", and they were mathematical geniuses who took all maths courses and got "every question right".

    In summary, I would avoid trendy courses like this. Professors, in general, don't take students on such courses seriously. (I ended up doing well in research, but in computer science -- the field anyone could get into with a science degree of any kind a couple of decades ago!)
     
  14. Jul 26, 2009 #13
    That wasn't so much a comment intended to indicate genuine non-understanding, but rather to indicate the buzz-word connotations behind the term, and hence its dilution as a real concept.
     
  15. Jul 26, 2009 #14
    I am inclined to disagree with the comparison between my proposed line of study and your combined degree. In particular, I am working towards something very specific, and I have a fairly good sense of direction.
     
  16. Jul 26, 2009 #15
    Wow, I write like a pretentious jerk!
     
  17. Jul 26, 2009 #16
    For QM, I actually started with Shankar, which took some doing, but I have a good grasp of the ideas. I'm supplementing it with Bohm and Landau/Lifgarbagez when explanations or mathematics are lacking, respectively. Cohen-Tannoudji has been on the way from India for some time now (heh), and I will probably look at Sakurai after I take some more advanced maths.

    I've taken the four calc courses, and one lin-alg course. I also feel that I could breeze through a second lin-alg course on transformations, because, due to my QM course, I've been reading Friedberg/Insel/Spence Lin-Alg, as well as Dover copies of Shilov Lin-Alg and Byron/Fuller Math of QM. I've also read a bit of Debnath/Mikusinski Hilbert Spaces (although it's dry), and, as silly as it sounds, have spent days on Wikipedia absorbing a mish-mash of geometrical concepts like metric spaces, affine spaces, coordinate spaces, manifolds, bundles, topology, etc.

    I also dipped into multilinear algebra by touching on tensors and reading the "extraneous" generalizations to tensors in the vector calc topics, (also, in trying to wrap my mind around the interpretation of Einstein notation).

    I have Griffiths' electrodynamics, which I read pre-vector-calc, and I have a deep enough understanding of vector calc now that the physical concepts will probably seem very natural. I will be looking at Jackson in my later years, but I'm not certain I'll need it.

    I also have a lot of mechanics books that I'm sitting on, which I bought to help me understand the Hamiltonian. I've got Landau/Lifgarbagez, and Gelfand/Fomin (among many other Dovers).

    For me, the big Field of Uncertainty surrounds statistical mechanics and mathematical physics. I'll take a look at the books you've mentioned.

    Thanks again!
     
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