(Solved): 2 Operators, means and expectation values (5 marks) With small modifications to Exercise 9.2, we ma ...

2 Operators, means and expectation values (5 marks) With small modifications to Exercise 9.2, we may define an amplitude-modulated complex wave: \[ f(t)=a_{0} \exp \left(i \omega_{0} t\right)+\frac{\alpha}{2} a_{0}\left\{\exp \left(i\left[\omega_{0}+\omega_{1}\right] t\right)+\exp \left(i\left[\omega_{0}-\omega_{1}\right] t\right)\right\} \] whose (amplitude) frequency spectrum will be given by: \[ F(\omega)=\frac{a_{0}}{2}\left\{\delta\left(\omega_{0}-\omega\right)+\frac{\alpha}{2} \delta\left(\omega_{0}+\omega_{1}-\omega\right)+\frac{\alpha}{2} \delta\left(\omega_{0}-\omega_{1}-\omega\right)\right\} \] where, in contrast to Exercise 9.2, there are no corresponding terms for negative frequencies. Given that the power (or intensity) spectrum is proportional to the modulus-squared of the amplitude spectrum above, find the mean frequency of the power spectrum, defined by: \[ \bar{\omega}=\frac{\int_{0}^{\infty} \omega F^{*}(\omega) F(\omega) d \omega}{\int_{0}^{\infty} F^{*}(\omega) F(\omega) d \omega} \] (Note that you should not need to evaluate the integral of the Dirac \( \delta \)-function, or its square, as it occurs in both the numerator and denominator of Equation 10.4. You may also find helpful the observations provided in the appendix to this problem sheet.] 1 Problem sheet 10 Show that the same result is obtained by calculating the expectation value of the operator, \( \hat{\boldsymbol{\omega}}=-i(d / d t) \) : \[ <\hat{\boldsymbol{\omega}}>=\frac{\int_{-\infty}^{\infty} f^{*}(t) \hat{\boldsymbol{\omega}} f(t) d t}{\int_{-\infty}^{\infty} f^{*}(t) f(t) d t} \]