Functional Derivates

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

2. For states $\vert\Psi\rangle$ corresponding to wave functions $\Psi({\bf r})$, the energy functional is a `function of a (wave) function'. Minimising $E[\Psi]$ 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 $f(x)$ is defined as

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

( $\delta x$ is a small deviation around the variable $x$).

Definition: The functional derivative of a functional $F[\Psi]$ is defined as

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

( $\delta \Psi$ is a small deviation around the function $\Psi$).

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

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

$$= \int d{\bf r} \Psi^*({\bf r}) \hat{H} \Psi({\bf r}) + \varepsilon... ...) \hat{H} \Psi({\bf r}) + \Psi^* ({\bf r}) \hat{H} \delta \Psi({\bf r}) \right]$$

$$+ \varepsilon^2 \int d{\bf r}\delta \Psi^* ({\bf r}) \hat{H} \delta\Psi({\bf r})$$ (3.6)

and therefore

$$\frac{\delta E[\Psi]}{\delta \Psi} = \int d{\bf r} \left[ \delta ... ...) \hat{H} \Psi({\bf r}) + \Psi^* ({\bf r}) \hat{H} \delta \Psi({\bf r}) \right]$$

$$\equiv \langle \delta \Psi\vert \hat{H}\vert \Psi \rangle + \langle \Psi\vert \hat{H}\vert \delta \Psi \rangle.$$ (3.7)