Lecture 3 | Modern Physics: Quantum Mechanics (Stanford)

One Response to “Lecture 3 | Modern Physics: Quantum Mechanics (Stanford)”

  1. admin Says:
    April 17th, 2009 at 11:25 pm

    A linear operator acts on a vector to give you a new vector. (Think of operators as matrixes)

    K |A> = |C>

    This together with the notion of a dual vector space and an inner product allows you to define what are called matrix elements of operators. These are a collection of numbers where each number represents a pair of states.

    The inner product of the K operator acting on the A vector and vector B : 

    ]

    is written as

    or Ksubab and is called the matrix element of K between vector B and vector A.

    To get a component of an abstract vector |A>:

    will give you the n th component.

    writing |A> in terms of its components:

    = sum over m of K sub nm A sub m.

    Linear operators can be multiplied together, the order matters. i.e KL acting on |A> is L acting on |A> to give a new vector, then K acting on it to give another.

    20:00

    Hermitian operators: Using H here but note this does not mean hamiltonian)

    they are defined such that: when sandwiched between two elements of a state pair = * (i.e to get the complex conjugate you don’t need to conjugate H).

    Taking it sandwiched between the same state:
     = *
    a hermitian operator can be defined as an operator which when sandwiched between the same state gives something real.
    Therefore its reflection in the x axis is the same i.e. it lies on it, is a conjugate of itself and is real.

    i.e Hsubab H*subba

    ****

    26:00

    Eigenvalues and Eigenvectors of operators.

    In general when you apply an operator to a vector you change its direction. For Hermitian operators there are vectors (directions) which if you apply the Hermitian operator you wont change the direction. These are called Eigenvectors.

    If we have a hermitian operator H which acts on an eigenvector lambda (using l since i don’t have greek symbols loaded) it will scale it by a number (lets also call that lambda, called the eigenvalue), i.e.:

    H|L> = L|L>

    Theorem 1: All of the Eigenvalues of a Hermitian operator are real.

    Theorem 2.: If there are different Eigenvalues of a particular Hermitian operator then the Eigenvectors are orthogonal.

    Theorem 3.: There are the same number of eigenvectors of hermitian operators as the number of dimensions.

    This means that the eigenvectors of hermitian operators form base vectors.

    *********

    40:00

    The postulates of QM

    1. States correspond to the collection of vectors.
    2. Observables correspond to the collection of hermitian operators.
    3. The values of an observable are the eigenvalues. **(the eigenvalues are real numbers (surely they are prime numbers because the operators could have been scaled to make them unit vectors, and all non primes are scaled versions of primes - this then makes sense why the zero of the zeta function correspond to primes)).
    4. The states for which the observables are certain are the eigenvectors of that observable.
    5. If you choose a state which is not an eigenvector of a particular observable then the measurable along the axis of the eigenvector(Lsubm|A>) when squared or multiplied by its complex conjugate is the probability of finding that measurements eigenvalue. (you have to square since its a complex number and probabilities are real).

    ****
    up to 1hr.

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