Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Harmonic Oscillator equivalence

  1. Feb 6, 2017 #1
    Hello, I'm studying the section 2.2 of "Introduction to Quantum Mechanics, 2nd edition" (Griffiths), and he shows this equation $$\frac{\partial^2\psi}{\partial x^2} = -k^2\psi , $$ where psi is a function only of x (this equation was derivated from the time-independent Schrödinger equation) and k is defined as $$\frac{\sqrt{2mE}}{\hbar}.$$

    Then, he refers to the equation above as being the classical simple harmonic oscillator equation. That's where my question comes: I can't exactly see how he made this connection with the following harmonic oscillator equation (for masses and springs with constant k, for example) $$ m\frac{d^2 x}{dt^2} = -kx$$

    Any help will be very appreciated. Thanks :)
     
  2. jcsd
  3. Feb 6, 2017 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Mathematically it is the same equation, just changing the names of the variables.
     
  4. Feb 7, 2017 #3

    Demystifier

    User Avatar
    Science Advisor

    Which is a misleading coincidence.
     
  5. Feb 7, 2017 #4

    Demystifier

    User Avatar
    Science Advisor

    The ##k## in the first equation (Sec. 2.2) has nothing to do with the ##k## in the last equation (Sec. 2.3). Your first and second equation have nothing to do with harmonic oscillator, and Griffiths does not say that they do.
     
  6. Feb 7, 2017 #5

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    I do not see why you want to claim that my post is misleading. It is the same differential equation. That it describes completely different things that a priori are physically different is a different matter. (Also, one comes with boundary conditions at two points and the other with initial conditions in a single point, which makes them mathematically different. But mathematically, the differential equations themselves are the same.)
     
  7. Feb 7, 2017 #6

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    It's interesting how easy it is to give a misleading impression. Why didn't Griffiths just say "and we have a well-known second-order ODE, whose solution is ..."? Why did he have to mention the classical SHO at all? I'll bet he never even thought about it. It was just a way to say "here's an equation we already know how to solve."
     
  8. Feb 7, 2017 #7

    Paul Colby

    User Avatar
    Gold Member

    Boundary conditions and normalization factors? For finding the set of possible solutions certainly. Much of the physics is often elsewhere.
     
  9. Feb 7, 2017 #8

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    It's NOT a coincidence, once you realize you don't need Schroedinger wavefunctions, but can do very well with Heisenberg matrices (or, if you prefer, time-dependent operators in the Heisenberg picture).
     
  10. Feb 7, 2017 #9
    I was thinking about that; actually, it's the answer that makes more sense to me. I just thought that was kind of "weird" the similarity of the equations on a physical context, but if the focus is the "shape" of the equation (Mathematically), makes all sense. Thank you :)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Harmonic Oscillator equivalence
  1. Harmonic Oscillator (Replies: 2)

  2. Harmonic oscillator (Replies: 9)

  3. Harmonic Oscillator (Replies: 1)

  4. Harmonic oscillator (Replies: 1)

  5. Harmonic oscillator (Replies: 1)

Loading...