assume a quantum particle in 1d subjected to the potential defined as

Question

Assume a quantum particle in 1D subjected to the potential defined as V(x)=\left\{\begin{array}{lll}
\infty & \text { for } & x<-3 \\
0 & \text { for } & -3<x<3 \\
\infty & \text { for } & 3<x
\end{array}\right. and the particle is prepared in the quantum state \Psi(x)=A\left[16-(x-\sqrt{5})^{2}(x+\sqrt{5})^{2}\right], \quad \text { for } \quad-3 \leq x \leq 3 where A is a real and positive constant. 1.1 Write the time-dependent Schrödinger equation in the three regions naturally separated. 1.2 Why is it enough to describe the wavefunction in the region -3 < x < 3? 1.3 Normalise the wavefunction (1); namely, calculate A. 1.4 Obtain | (x)|² and use Julia to sketch it. 5 What is the probaility to find the particle in the region -3 < x<-1? 1.6 What is the probaility to find the particle in the region -1 < x < 1?(1 r 1.7 What is the probaility to find the particle in the region 1 < x < 3? (*need to do any further integrals!) 1.8 What is the probaility to find the particle in the region -3 < x < 0? 1.9 What is the probaility to find the particle in the region 0 <x < 3? 1.10 What is (x)? You can obtain this value using geometric arguments or integral.doing(1 mark)an 1.11 Obtain (x2), (p) and (p²). 1.12 Use the results in the previous two items to show that the variance on the position and momentum of this quantum particle are given by(2 marks) \sigma_{x}^{2}=\left\langle x^{2}\right\rangle-\langle x\rangle^{2}=\frac{16325}{8019} \quad \text { and } \quad \sigma_{p}^{2}=\left\langle p^{2}\right\rangle-\langle p\rangle^{2}=\frac{295 \hbar^{2}}{243}