Part 11, 13, 14
Outline, Schedule

PHYS 424 Notes

Part 13

  1. Chapter 3
    1. Statistical Interpretation

      Postulates, modified from Griffiths:

      1. The state of a particle is represented by a normalized vector in the Hilbert Space L2.

        yn(x) = Ö(2/a) sin(npx/a)

      2. Observable quantities are represented by Hermitian operators.

        x, p = - i hbar d/dx , T = p2/(2m), V = mw2/2, H = T + V, ...

      3. The expected value of an observable Q is <y | Q | y>.

      4. If you measure an observable Q on a particle in the state |y>, you are certain to get one of the eigenvalues of Q.

        En = n2p2 hbar2 / (2 m a2)

      5. The probability of getting the particular eigenvalue l is equal to the absolute square of the l component of |y>, when expanded in the orthonormal basis of eigenvectors.

        If y = Sn anyn  , <H> = Sn |an|2 En

      6. A measurement of the observable Q on a particle in the state |y> is certain to return the value l if and only if |y> is an eigenvector of Q with eigenvalue l.

        Need an = dnn'

      Discrete vs continuous eigenvalues

        Discrete Continuous
      Eigenvalue equation Q|n>=ln|n> Q|k>=lk|k>
      Orthonormality <m|n>=dmn <l|k>=d(l-k)
      Expansions |y>=Sn cn|n> |y>=òdk c(k)|k>
      Coefficients cn=<n|y> c(k)=<k|y>
      Probability of getting a particular eigenvalue |cn|2 |c(k)|2dk

      Coordinate space vs momentum space

      yn(x) = <x|n>

      fn(p) = <p|n> = <p|x><x|n> = òdx y*p(x) yn(x)

      Operators imply matrices

      If H y = E y,

      you can expand y in terms of any complete states (i.e. basis): y = Sn cn yn to get the equation

      Sn cn H |n> = E Sn cn |n>

      Taking the inner product with <m| [in other words, multiplying by ym* and integrating]

      Sn <m|H|n> Cn = E dmn Cn

      which is a matrix eigenvalue equation. Hence solving the Schroedinger equation is equivalent to diagonalizing the Hamiltonian matrix in an arbitrary basis. This is why solving the Schroedinger equation is often referred to as "diagonalizing the Hamiltonian."

Last Revised 01/10/05

Part 11, 13, 14
Outline, Schedule