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B What are operators?

  1. Jul 8, 2017 #1
    I'm a beginner in QFT, starting out with D. J. Griffiths' book in this topic.

    I have a question on the operators used in QM. What are operators? What is the physical significance of operators? I can understand that ##\frac {d}{dt}## to be an operator, but how can there be a total energy operator, a potential energy operator, and what is the meaning of subtracting them to give a Hamiltonian? (##\hat H = \hat T - \hat V##)
     
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  3. Jul 8, 2017 #2
    The Hamiltonian is the sum of potential and kinetic energy ##\hat{H}=\hat{T}+\hat{V}## not the difference. The Hamiltonian operator can be said to act as the total energy operator. This means that the eigenvalues of the Hamiltonian are the total energies ##E_{n}## corresponding to a given eigenstate ##\phi(\mathbf{x})_{n}## as
    $$\hat{H}\phi(\mathbf{x})_{n}=E_{n}\phi(\mathbf{x})_{n}$$.
     
  4. Jul 8, 2017 #3
    What are eigenvalues and eigenstate?
     
  5. Jul 8, 2017 #4
    Are you familiar with linear algebra? A Vector ##\mathbf{x}## which has the property that when multiplied by a matrix ##\mathbf{A}## as
    $$\mathbf{A}\mathbf{x}=\alpha\mathbf{x}$$
    produces the same vector ##\mathbf{x}## multiplied by a scalar ##\alpha## is called an eigenvector. The scalar ##\alpha## is called an eigenvalue. This is analogous to the time independent Schrödinger equation
    $$\hat{H}\phi(\mathbf{x})_{n}=E_{n}\phi(\mathbf{x})_{n}$$
    where ##E_{n}## is an eigenvalue and ##\phi(\mathbf{x})_{n}## is an eigenstate or eigenfunction.
     
  6. Jul 8, 2017 #5
    Understood about eigenvalues. But what does an operator physically signify? What is the physical significance of energy operators?
     
    Last edited: Jul 8, 2017
  7. Jul 8, 2017 #6

    vanhees71

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    Have you studied classical mechanics in the Hamilton formalism, including Poisson brackets and symmetries (Noether)? If not, chances are pretty bad that you understand what's behind the formalities of QT. So it's wise, to study classical physics to some level of sophistication first. This holds the more for classical electrodynamics and relativistic quantum field theory. Also here it's important to study the classical theory first, including a firm understanding of the relativistically covariant formulation and some representation theory of the Lorentz and Poincare group.
     
  8. Jul 8, 2017 #7
    It's true that I didn't study classical formalism, because I'm somehow more interested in quantum. Can you recommend a book, or website, that can give me a overview of these and form a concept with which I can understand at least preliminary quantum mechanics?
     
  9. Jul 8, 2017 #8
    I used Classical Dynamics of Particles and Systems by Stephen T. Thornton & Jerry B. Marion. I think it's pretty good for an introductory mechanics text.
     
  10. Jul 8, 2017 #9

    Stephen Tashi

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  11. Jul 8, 2017 #10

    Vanadium 50

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    There are lots of people who come here saying "I'm not interested in studying the middle. Take me straight to the end!" They tend not to be successful.
     
  12. Jul 8, 2017 #11
    I agree. I was really interested in quantum, and knew a lot of maths, but I simply couldn't understand it properly because I had no knowledge in classical physics. So I decided to study physics at university, I've completed my first year (lot of mechanics, EM, more maths) and even though I haven't started quantum properly yet, I'm more confident now that I'll be better at it.
     
  13. Jul 9, 2017 #12
    To whom do you intend to address this to?
     
  14. Jul 9, 2017 #13
    Clearly to you, you haven't done classical mechanics and based on your posts you haven't done much linear algebra either. Believe me I'm not interested in classical mechanics either, but to do quantum properly you have to start from the beginning, and go through the steps. Of course there will be people saying classical mechanics is not necessary, and I suppose you can learn quantum without it, but having completed 1 year of undergraduate physics I'm so grateful to myself that I started from the beginning.
     
  15. Jul 9, 2017 #14

    Vanadium 50

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    To anyone who wants to go straight to one part of physics without covering the intervening material. For example, someone who wants to understand QM and only QM without any classical physics.
     
  16. Jul 10, 2017 #15
    I was wrong to ask this question. :bow::headbang::oldgrumpy:
     
  17. Jul 10, 2017 #16

    Vanadium 50

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    The question isn't wrong. The plan is wrong. You can't get to the end without going through the middle.
     
  18. Jul 10, 2017 #17

    vanhees71

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    Yes, and the answers are very valuable, although maybe the OP doesn't realize it :-(.
     
  19. Jul 10, 2017 #18
    Anyways, I have found a book in which one can start even without knowing classical formalism. It acts as a training book and bridges the gap for one who does not know classical. I spoke with the author, described my situation, and he said that I should read that book before starting out with Griffiths. I've switched over to that book for the time being.
     
  20. Jul 10, 2017 #19
    May I ask what the name of said book is? It is true that one can do quantum mechanics to some extent without having seen either the Lagrangian nor the Hamiltonian formulation of quantum mechanics: engineers doing introductory courses on nanotechnology do it, but it does require some acceptance, i.e. accepting that one will not be able to truly understand the origin of many formulae. Having said this I do think that one should pursue the study of the things that one finds interesting. Maybe on the way you will interest yourself in classical mechanics and will try to understand them as well in order to more fully understand what you are reading.
    So I encourage you to read as many physics as you can, but know that the way will be long.

    Also knowledge of linear algebra and analysis is recommended.
     
  21. Jul 12, 2017 #20

    ftr

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    IMHO always check Google first so you can get some info before you formulate your question.

    https://en.wikipedia.org/wiki/Operator_(physics)
     
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