slides6-cutout.html

      <div class="slide build-focus-visible" group="theme2">
        <h1>Recap</h1>
        <div>Scalar product spaces.</div>
        <ul>
          <li step="1">Vector space equipped with scalar product.
          <li step="2">Why not always deal in orthonormal bases? - In QM we will.
          <li step="3,5">Covector space basis can be defined as
            $$\xvec{e}^i(\xvec{e}_j) = \delta^i_j$$ <span step="5">$N$
            equations per basis vector.</span>
          <li step="4,5">Scalar product gives mapping of vectors and covectors
            $$\xvec{f}(\xvec{x}) := \xvec{f} \cdot \xvec{x}$$ <span step="5">One
            equation per vector.</span>
        </ul>
      </div>


      <div class="slide build-focus-visible" group="theme2">
        <h1>Dirac Notation</h1>
        <div>Used in quantum mechanics.</div>
        <ul>
          <li step="0">Vector space in QM is functions that are solutions to the
            Schr&ouml;dinger equation.
          <li step="1">Complex valued time dependent or time independent
            functions of 3d space (aka, fields) form a vector space. (Check
            axioms.)
          <li step="2">Have a natural scalar product. (Check axioms.)
          <li step="3">Schr&ouml;dinger operator is linear, so solutions of the
            equation for one eigenvalue (next session) form a subspace.
          <li step="4">In QM, every vector worth mentioning is the eigenvector
            of some operator.
          <li step="5">Idea in Dirac notation is to always label vectors with
            eigenvalues of some operator, and use those as "coordinates".
        </ul>
      </div>

      <div class="slide build-focus-visible continued" group="theme2">
        <h1>Dirac Notation</h1>
        <table>
          <tr>
            <th>
            <th style="text-align:left">Dirac notation
            <th style="text-align:left">Einstein notation
          </tr>
          <tr step="1">
            <td>Any vector
            <td>$\ket{\psi} \in V$
            <td>$\xvec{x} \in V$
          </tr>
          <tr step="2">
            <td>Vector basis
            <td>$\sum_i \psi_i \ket{i}$
            <td>$x^i \xvec{e}_i$
          </tr>
          <tr step="3">
            <td>Vector coordninates
            <td>$\psi_i = \braket{i|\psi}$
            <td>$x^i \xvec{e}_i$
          </tr>
        </table>
      </div>

      <div class="slide build-focus-visible continued" group="theme2">
        <h1>Dirac Notation</h1>
        <table>
          <tr>
            <th>
            <th step="1,2,3" style="text-align:left">Dirac notation
            <th step="1,2" style="text-align:left">Einstein notation
          </tr>
          <tr>
            <td step="1,3">Covector applied to vector
            <td step="1,3">$\braket{\phi|\psi} = \sum_{ij} \phi_i^\ast \psi_j \braket{i|j}
              = \sum_i \phi_i^\ast \psi_i$
            <td step="1">$\xvec{x} (\xvec{y}) = x_i y^j \xvec{e}^i(\xvec{e}_j)$
          </tr>
          <tr>
            <td step="2,3">Scalar product
            <td step="2,3">$\braket{\phi|\psi} = \sum_{ij} \phi_i^\ast \psi_j \braket{i|j}
              = \sum_i \phi_i^\ast \psi_i$
            <td step="2">$\xvec{x} \cdot \xvec{y} = x^i y^j \xvec{e}_i \cdot \xvec{e}_j = x^i y^j g_{ij}$
          </tr>
        </table>
      </div>

      <div class="slide build-focus-visible continued" group="theme2">
        <h1>Dirac Notation</h1>
        <table>
          <tr>
            <th>
            <th style="text-align:left">Dirac notation
            <th style="text-align:left">Einstein notation
          </tr>
          <tr>
            <td step="1">Operators
            <td step="1">$\ket{\psi}^\prime = A \ket{\psi}$
            <td step="1">$\xvec{x}^\prime = A \xvec{x}$
          </tr>
        </table>
      </div>

      <div class="slide build-focus-visible continued" group="theme2">
        <h1>Dirac Notation</h1>
        <div>Operator coordinates</div>
        <ul>
          <li step="1">$\ket{\psi}^\prime = A \ket{\psi}$
          <li step="2">$\psi^\prime_i = \sum_{j} A_{ij} \psi_j$
          <li step="3">$A_{ij} = \braket{i|A|j}$
          <li step="4">How can we see this:
          <li step="5">$\ket{\psi}^\prime = A \ket{\psi}$
          <li step="6">$\ket{\psi}^\prime = \sum_j A \ket{j} \braket{j|\psi}$
          <li step="7">$\braket{i|\psi}^\prime = \bra{i} \sum_j A \ket{j} \braket{j|\psi}$
          <li step="8">$\braket{i|\psi}^\prime = \sum_j \bra{i} A \ket{j} \braket{j|\psi}$
          <li step="9">$\psi^\prime_j = \sum_{i} A_{ij} \psi_i$
        </ul>
      </div>