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Functional Derivates

1. If the Hilbert space belonging to \bgroup\color{col1}$ \hat{H}$\egroup was finite dimensional (for example in the case of the two-level system), the energy functional would just be a quadratic form and \bgroup\color{col1}$ \Psi=(c_1,c_2)^T$\egroup would just be a two-component vector.

2. For states \bgroup\color{col1}$ \vert\Psi\rangle$\egroup corresponding to wave functions \bgroup\color{col1}$ \Psi({\bf r})$\egroup, the energy functional is a `function of a (wave) function'. Minimising \bgroup\color{col1}$ E[\Psi]$\egroup means that we have to set its first functional `derivative' to zero (in very much the same way as we set the first derivative of a function to zero in order to find its minimum).

Definition: The derivative of a function \bgroup\color{col1}$ f(x)$\egroup is defined as

$\displaystyle \frac{d f (x) }{d x} \equiv
\lim_{\varepsilon\to 0} \frac{ f[x+\varepsilon \cdot \delta x] -f[x]}{\varepsilon}.$     (3.4)

( \bgroup\color{col1}$ \delta x$\egroup is a small deviation around the variable \bgroup\color{col1}$ x$\egroup).

Definition: The functional derivative of a functional \bgroup\color{col1}$ F[\Psi]$\egroup is defined as

$\displaystyle \frac{\delta F[\Psi]}{\delta \Psi}\equiv
\lim_{\varepsilon\to 0} \frac{ F[\Psi+\varepsilon \cdot \delta \Psi]-F[\Psi] }{\varepsilon}.$     (3.5)

( \bgroup\color{col1}$ \delta \Psi$\egroup is a small deviation around the function \bgroup\color{col1}$ \Psi$\egroup).

So we recognise that everything is really quite analogous to ordinary derivative. The functional derivative of \bgroup\color{col1}$ E[\Psi]$\egroup is obtained from calculating

    $\displaystyle E[\Psi+\varepsilon \cdot \delta \Psi] =
\int d{\bf r} \left\{ \Ps...
...
\hat{H} \left\{ \Psi({\bf r}) + \varepsilon \cdot \delta \Psi({\bf r})\right\}$  
  $\displaystyle =$ $\displaystyle \int d{\bf r} \Psi^*({\bf r}) \hat{H} \Psi({\bf r})
+ \varepsilon...
...) \hat{H} \Psi({\bf r})
+ \Psi^* ({\bf r}) \hat{H} \delta \Psi({\bf r}) \right]$  
  $\displaystyle +$ $\displaystyle \varepsilon^2 \int d{\bf r}\delta \Psi^* ({\bf r}) \hat{H} \delta\Psi({\bf r})$ (3.6)

and therefore
    $\displaystyle \frac{\delta E[\Psi]}{\delta \Psi} =
\int d{\bf r} \left[ \delta ...
...) \hat{H} \Psi({\bf r})
+ \Psi^* ({\bf r}) \hat{H} \delta \Psi({\bf r}) \right]$  
  $\displaystyle \equiv$ $\displaystyle \langle \delta \Psi\vert \hat{H}\vert \Psi \rangle +
\langle \Psi\vert \hat{H}\vert \delta \Psi \rangle.$ (3.7)


next up previous contents index
Next: Lagrange Multiplier Up: The Variational Principle Previous: The Variational Principle   Contents   Index
Tobias Brandes 2005-04-26