The state of a harmonic oscillator is given by \(\Psi = \frac{1}{\sqrt2}\psi_0 + \frac{1}{\sqrt{3}}\psi_1 - \frac{1}{\sqrt{6}}\psi_2\) , where\( \psi_0, \psi_1\), and\( \psi_2 \)are normalized wave functions of the ground, first excited, and second excited states, respectively. Which of the following statements are true?
1
A measurement of the energy of the system yields \(E = \frac{1}{2} \hbar \omega\) with non zero probability.
2
A measurement of the energy of the system yields \(E = \frac{5}{2} \hbar \omega\) with non zero probability.
3
Expectation value of the energy of the system \(\langle E \rangle = \frac{7}{12} \hbar \omega \) .
4
Expectation value of the energy of the system \( \langle E \rangle = \frac{3}{2} \hbar \omega\) .