**Quantum Mechanics Problem: Particle in a 1D Box****Problem Description:**A particle of mass \( m \) is confined in a one-dimensional box of length \( a \), constrained within the range \( 0 < x < a \). After a measurement, where the system interacts with a device, the system's final state is represented by:\[\psi(x, t = 0^+) = c \left( e^{-i\alpha}\phi_2(x) + \sqrt{2} e^{-i\delta} \phi_3(x) - \phi_4(x) \right)\]where \(\{\phi_n\}\) is a complete set of orthonormal wave functions solving the Time-Independent Schrödinger Equation (TISE), and \(\alpha = \frac{\pi}{4}\) and \(\delta = \frac{\pi}{3}\).If extensive experiments are executed to measure energy levels in prepared systems, it's possible to identify statistical properties of potential outcomes. Address the following questions. Part (e) requires careful attention and is time-consuming.1. **Integral Calculation:** What does this integral \(\int_0^a \phi_k^*(x) \phi_n(x) \, dx\) equal to?2. **Normalization Constant:** What is the normalization constant?3. **Probability Measurement:** What is the probability for the particle to be measured in energy state \( n \)?4. **Predicted Average Energy:** What is the predicted average energy from numerous energy measurements?**[Note: Graphs or diagrams are not present in the document.]** - What is the predicted standard deviation in the energy measurements?

Given: The final state of the system is specified as follows:…

Answered: 4. A particle of mass, m, is placed in…

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