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courses:lecture:pplec:pplecnwellsys 2011/08/10 16:11 courses:lecture:pplec:pplecnwellsys 2011/08/26 10:44 current
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\vdots &  & \ddots & \ddots & \ddots \\ \end{array}\right] \; \; . $$ \vdots &  & \ddots & \ddots & \ddots \\ \end{array}\right] \; \; . $$
-**//Our goal://** Find both the possible eigenstates of the electron in the potential landscape and the energy of each eigenstate.  +**//Our goal://** Find the possible eigenstates of the electron in the potential landscape and the energy of each eigenstate. 
  * To do this, let's again use the energy eigenvalue equation   * To do this, let's again use the energy eigenvalue equation
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\vdots \\ \end{array}\right] \; \; .$$ \vdots \\ \end{array}\right] \; \; .$$
-Performing the matrix algebra, we find that at the ends of the vector, $E=\alpha + \beta$.  But, for every other entry in the resulting expression, $E=\alpha + 2\beta$.  So, if this is a very long chain of potential wells, it is //close// to being an eigenstate.  Would a vector with alternating values of +1 and -1 satisfy the eigenvalue equation?+Performing the matrix algebra, we find that at the ends of the vector, $E=\alpha + \beta$.  For every other entry in the resulting expression,however, $E=\alpha + 2\beta$.  If this is a very long chain of potential wells, a state with $c_{i}=1$ is //close// to being an eigenstate.  Would a vector with alternating values of +1 and -1 satisfy the eigenvalue equation?
-  * Guessing any further for eigenvectors is going to become very difficult, so let's see what we can find about the eigenenergies.  Let's again look at the most general expression for the energy eigenvalue equation.+  * Guessing any further for eigenvectors is going to become very difficult, so let's see what we can find about the eigenenergies instead.  Let's again look at the most general expression for the energy eigenvalue equation.
$$ \left[\begin{array}{ccccccc} $$ \left[\begin{array}{ccccccc}
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Let's graph the the as a function of $k$. Let's graph the the as a function of $k$.
-Add image: pplecnwellsys1+{{courses:lecture:pplec:pplecnwellsysfig1.png|Add image: pplecnwellsys1}}
-Notice that, no matter how long the chain of potential wells is, the spread in energies never exceeds $4\beta$.+Recall that $\beta$ has a negative value; this is very important for graphing the energy correctly.  Also notice that, no matter how long the chain of potential wells is, the spread in energies never exceeds $4\beta$.
  * Finally, since we know that each coefficient for an eigenstate is determined by an envelope function, we can write that the energy eigenstates corresponding to a particular wave vector $k$ are   * Finally, since we know that each coefficient for an eigenstate is determined by an envelope function, we can write that the energy eigenstates corresponding to a particular wave vector $k$ are
-$$\psi_{k}(x) = \sum_{p=1}^{n}A \sin{kpa} \phi(x-pa) \; \; . $$+$$\psi_{k}(x) = \sum_{p=1}^{n}A \sin{(ikpa)} \; \phi(x-pa) \; \; . $$ 
 + 
 +This function is so important, it has it's own name; it is called a Linear //Combination of Atomic Orbitals// (LCAO).
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