Exercise 8.1

A linear operator \(\mathrm{O}\) has the following two properties: \[ \mathrm{O}(x+y) = \mathrm{O}(x) + \mathrm{O}(y) \]

\[ \mathrm{O}(cx) = c \, \mathrm{O}(x) \]

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The operator \(\mathrm{X}\) is defined by: \[ \mathrm{X} \psi(x) = x \psi(x) \]

\[ \begin{flalign*} \mathrm{X}(\psi(x) + \phi(x)) &= x(\psi(x) + \phi(x))&\\ &= x\psi(x) + x\phi(x)\\ &= \mathrm{X}(\psi(x)) + \mathrm{X}(\phi(x)) \end{flalign*} \]

\[ \begin{flalign*} \mathrm{X}(c\psi(x)) &= x c\psi(x)\\ &= c x\psi(x)\\ &= c \, \mathrm{X}(\psi(x)) \end{flalign*} \]

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The differentiation operator \(\mathrm{D}\) is given by: \[ \mathrm{D} \psi(x) = \frac{\mathrm{d}\psi(x)}{\mathrm{d}x} \]

Using the well known rules of differentiation (see also Exercise 2.4 in Classical Mechanics) we get: \[ \begin{flalign*} \mathrm{D} (\psi(x) + \phi(x)) &= \frac{\mathrm{d}(\psi(x) + \phi(x))}{\mathrm{d}x}&\\ &= \frac{\mathrm{d}\psi(x)}{\mathrm{d}x} + \frac{\mathrm{d}\phi(x)}{\mathrm{d}x}\\ &= \mathrm{D} \psi(x) + \mathrm{D} \phi(x) \end{flalign*} \]

\[ \begin{flalign*} \mathrm{D} (c \, \psi(x)) &= \frac{\mathrm{d}(c \, \psi(x))}{\mathrm{d}x}&\\ &=c \frac{\mathrm{d}\psi(x)}{\mathrm{d}x}\\ &= c \, \mathrm{D} \psi(x) \end{flalign*} \]