A particle of mass, \( m \), moves freely inside an infinite potential well spanning the range, \( 0 < x < b \). Somehow, we know that the initial wave function for the particle at \( t = 0 \) is given as:\[\psi(x, 0) = \sqrt{\frac{2}{7}} \phi_1(x) - \frac{\sqrt{3}}{\sqrt{7}} \phi_2(x) + \frac{\sqrt{9}}{\sqrt{7}} \phi_3(x) - \frac{\sqrt{2}}{\sqrt{7}} \phi_4(x)\]**Problems:**d) At \( t = 0 \), what are the energies that might get measured, and what are the probabilities?e) At \( t = \frac{1000 m b^2}{h \pi^2} \), what is the average energy?f) At \( t = \frac{1000 m b^2}{h \pi^2} \), what is the second moment of the energy?g) Calculate the average energy of the system as a function of time.h) If the energy of the system is measured to be \( \frac{9 \pi^2 \hbar^2}{2 m b^2} \) at \( t = 0 \), what will \( \psi(x, t) \) equal just after the measurement?i) If you then measure the energy of the system again at \( t = \frac{1000 m b^2}{h \pi^2} \), what are the energies that might get measured, and what are the probabilities?

In quantum mechanics, a particle in a box of length b can be described by the wave function: ϕn =…

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A particle of mass, m, moves freely inside an infinite potential well spanning the range, 0