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Consider a quantum mechanical particle of mass m in a state described by the normalised wave function \Psi(x, t)=A e^{-a\left(\left(m x^{2} / \hbar\right)+i t\right)} where a and A are a positive real constants. (a) Determine an expression for the probability density function P(x, t) and sketch itsshape. (b) Explain what is meant by normalisation in this context and show that A=\left(\frac{2 a m}{\pi \hbar}\right)^{1 / 4} ) Show that the expectation value for particle position (x) = 0 and the expectation value for position squared (x²) = ħ/4am. .) For what potential energy function V(x) does V(x, t) satisfy the time-dependent Schrödinger Equation? You may find the following results useful: \int_{-\infty}^{\infty} e^{-\alpha x^{2}} d x=\sqrt{\frac{\pi}{\alpha}} \int_{0}^{\infty} x^{2} e^{-\alpha x^{2}} d x=\frac{1}{4 \alpha} \sqrt{\frac{\pi}{\alpha}}

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