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The adjoint of an operator$\widehat{\mathrm{Q}}$ is defined as the operator${\widehat{Q}}^{鈥�}$ such that:$鉄�\mathrm{f}鈭�\widehat{\mathrm{Q}}\mathrm{g}鉄�=鉄�{\widehat{\mathrm{Q}}}^{鈥�}\mathrm{f}鈭�\mathrm{g}鉄�$

(a)

The adjoint of an operator $\widehat{\mathrm{Q}}$is defined as the operator ${\widehat{\mathrm{Q}}}^{鈥�}$such that:

$鉄�\mathrm{f}鈭�\widehat{\mathrm{Q}}\mathrm{g}鉄�=鉄�{\widehat{\mathrm{Q}}}^{鈥�}\mathrm{f}鈭�\mathrm{g}鉄�$

For the position operator $\mathrm{x}$, we have:

For the imaginary number $i$, we have:

For the operator $\frac{\mathrm{d}}{\mathrm{dx}}$, we can use integration by parts to find:

role="math" localid="1655396772124"

b)

The Hermitian conjugate of the harmonic oscillator raising operator is

${\mathrm{a}}_{+}=\frac{1}{\sqrt{2\mathrm{鈩�}\mathrm{尘蝇}}}(鈭�\mathrm{ip}+\mathrm{尘蝇x})$

the momentum$p$ and the position$x$ operators Hermitian, that is ${\mathrm{x}}^{鈥�}=\mathrm{x}$, and ${\mathrm{p}}^{鈥�}=\mathrm{p}$, but the conjugate of the imaginary number is ${\mathrm{i}}^{鈥�}=鈭�\mathrm{i}$, thus :$\begin{array}{l}{\left({\mathrm{a}}_{+}\right)}^{鈥�}=\frac{1}{\sqrt{2\mathrm{鈩�}\mathrm{尘蝇}}}(\mathrm{ip}+\mathrm{尘蝇x})={\mathrm{a}}_{鈭�}\\ 鈬�{\mathrm{a}}_{+}={\mathrm{a}}_{鈭�}\end{array}$

c)

Solving the given equation as,

$鉄�f鈭�\left(\widehat{Q}\widehat{R}\right)g鉄�=鉄�{\widehat{Q}}^{鈥�}f鈭�\widehat{R}g鉄�$

Substituting the calculated conjugates in above equation as,

$\begin{array}{l}=鉄�{\widehat{R}}^{鈥�}{\widehat{Q}}^{鈥�}f鈭�g鉄�\\ =鉄�{\left(\widehat{Q}\widehat{R}\right)}^{鈥�}f鈭�g鉄�\end{array}$

Thus, the we can write: ${\left(\widehat{Q}\widehat{R}\right)}^{鈥�}={\widehat{R}}^{鈥�}{\widehat{Q}}^{鈥�}$.