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The use of Eigenvalues/function in QM

  1. Sep 21, 2007 #1
    Hello all. I'm in an introductory QM course as a physics major. As I understand it, one has an eigenfunction of an operator if one applies that operator to the function and gets a scaler multiple of it. Using the analogy of a rotation matrix, the eigenvectors represent the axis of rotation. I understand that if an operator doesn't change the vectors angle, it must be pointed along the axis of rotation.

    So what about the "direction" of eigenfunctions in QM? If the Hamiltonian operator just scales the function, and the eigenvalue is the energy, what sort of significance can be placed on the "direction" of the eigenfunction, as related to the operator? Why is the eigenvalue the energy? I guess I don't understand why eigenvalues/functions are used. What properties do they have that serve the math-needs of the problems at hand? Sorry for unclear questions, but I'm having problems verbalizing what I mean. Thanks in advance for the help.
  2. jcsd
  3. Sep 21, 2007 #2

    You're doing the right thing to look for analogies, but I think you've picked a misleading case here. The rotation matrix is a funny matrix. If we look at matrices in two dimensions, they generally stretch and distort the xy plane. But usually we can find two special directions where vectors get stretched without changing direction. These are the eigenvectors of the matrix. I'm pretty sure they are normally at right angles to one another. The point of identifying these special vectors is that if we represent other vectors in terms of these "base vectors", the matrix operation is especially simple.

    The rotation vector is funny because in two dimensions, NO vector gets mapped to itself. Unless you look at the complex-valued vectors (1,i) and (1,-i). These vectors actually do get mapped to multiples of themselves, and the eigenvalue is related to the angle of rotation. But they're not what we normally think of as vectors in the xy plane. Yes, in three dimensions the axis of rotation is, I suppose, an eigenvector; but it needs two more vectors to form a complete set of basis states. So it's a misleading example to use as your model of what eigenvectors and eigenvalues are good for.
  4. Sep 21, 2007 #3
    Eigenfunctions of the Hamiltonian [itex]H [/itex] are just position-space representations of eigenstates [itex]|\Psi_n \rangle [/itex]

    [tex] H |\Psi_n \rangle = E_n |\Psi_n \rangle [/tex]

    Note that eigenstates of [itex] H [/itex] have two important properties. First, the expectation value of energy in these states is [itex] E_n [/itex]

    [tex] \langle \Psi_n |H |\Psi_n \rangle = E_n [/tex]

    So, the energy measured in these states is [itex] E_n [/itex]. Moreover, there is no variance in measured energy values.

    Second, according to the non-stationary Schroedinger equation

    [tex] i \hbar \partial /\partial t |\Psi_n \rangle = H |\Psi_n \rangle = E_n |\Psi_n \rangle [/tex]

    the time dependence of eigenvectors is contained entirely in the (insignificant) phase factor

    [tex] | \Psi_n (t)\rangle = \exp(-\frac{i}{\hbar} E_n t)|\Psi_n (0)\rangle [/tex]

    which means that these states are, actually, independent on time. Stationary bound states of atoms and molecules are good examples of such eigenstates.

  5. Sep 22, 2007 #4
    I appreciate the replies. They are helpful, but I'm still not completely sure I understand. Would it be correct to say that the eigenstates are the only states that correspond to reality? By this I mean that when you make a measurement, the particle must be in an eigenstate, or a linear combination of eigenstates? So the "directions" sort of correspond to spatial realities? That's a pretty abstract concept, if correct. My follow-up question would be, what properties of the eigen-values/functions make this possible? The fact that it gives mutually perpendicular base-states? I've found that the math of quantum, as far as I can tell, seems to be built by necessity. For instance, their goal is to represent particles by waves, so in order to localize a wave, they construct a wave packet. They use Gaussian representations of the wave packet to make the square-integrals over all space convenient to calculate. These wave amplitudes, when squared, happened to agree with experiment if interpreted as probability intensities. So, following this line of thinking, I was trying to conceptualize the need eigen-equations served. I'm certainly getting closer to understanding, but wish my knowledge of Linear Algebra was stronger. Anybody have any links that explain the use of Linear Algebra in the context of QM? Thanks!
  6. Sep 22, 2007 #5
    No, eigenstates are not the only real states. For example, eigenstates of the Hamiltonian form an orthonormal basis of vectors in the Hilbert space. Any other vector (which also represents some legitimate state) is a linear combination of these eigenstates. And squares of absolute values of the coefficients in this linear combination are probabilities for measuring the corresponding energy values in this state.

    The connection to linear algebra (in Hilbert spaces) is the central mathematical idea of quantum mechanics. If your textbook doesn't explain this connection, I would suggest to find another textbook.

  7. Sep 22, 2007 #6
    Maybe the connection to linear algebra isn't the real problem. Are you quite sure you know why we use eigenfunctions when solving differential equations? Do you know the solutions for the Fourrier Equation (heat flow) for example on an infinite bar?
  8. Sep 23, 2007 #7
    Is it because the eigenfunctions give all the linearly independent possible solutions, in which all the solutions can be made up of? I guess that's the main point of their use: to give a basis set that any solution can be constructed from.
  9. Sep 23, 2007 #8
    There are lots of different basis sets you can use. But the eigenfunctions are special because they behave simply under the influence of the operation. Like the eigenvectors in linear algebra that keep their direction but only change their size.

    I think all the important analogies you need to draw from linear algebra will be found in the theory of differntial equations, especially cases like the Heat Equation I referred to above. Going to quantum mechanics only makes things more confusing because the straighforward analogies are all mixed up with questions like what is an operator and nonsense like the collapse of the wave function. Which isn't helpful at all to think about.

    Do you know why sine waves are the solution of the heat equation on an infinite bar?
  10. Sep 24, 2007 #9
    We skipped the chapter on Diff Eq. in Linear Algebra. I'll go back and look at the relevant sections. Thanks for the direction.
  11. Sep 24, 2007 #10
    Well, maybe I can give you an idea of what to look for. The eigenfunctions of the heat equation are sine waves. That means if you start with a sinusoidal distribution of temperature on a long bar, it STAYS sinusoidal with the passage of time. The shape
    of the distribution stays the same, it just decays and gets smaller. Different sine waves decay at different rates. So if you have a non-sinusoidal distribution, like a square wave or even a pulse, you just break it down into sine waves, follow the sine waves through time, and then add them up again.

    The sine waves that keep their shape are like the vectors that keep their direction. You shouldn't be looking for functions in QM that keep their DIRECTION, but functions that keep their shape.

    When a vector keeps its direction constant, the various component of that vector maintain the same ratio with respect to one another. That's exactly what the various points on a function do when it keeps it shape constant. That's the analogy you want.
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