Is My Approach to the Gaussian Wave Packet Problem Professional?

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Homework Statement
Problem involving normalization and Fourier transform
Relevant Equations
Wave function
hi,

I'm solving this statement,
1721513977862.png

1721513995402.png

We split into parts
1721514094971.png

1721514116323.png

1721514143924.png

Can some expert in Quantum say that my working is professional?
Kind wishes to you
 
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Parts (i) and (ii) are rigorously explained so one may say that they are professionally done. Part (iii) is not "professional", mainly because you make an assertion without justifying it. You say that "the probability density ##~| \Psi(x,t)|^2~## spreads over time indicating that the wave packet disperses as time progresses." To make the argument stick, you need to do the integral in equation (13), find ##~|\Psi(x,t)|^2~##, identify the width and argue that it is time-dependent.
 
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Thread 'Please tell me where I am going wrong in this integral'

##|\Psi|^2=\frac{1}{\sqrt{\pi b^2}}\exp(\frac{-(x-x_0)^2}{b^2}).## ##\braket{x}=\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dx\,x\exp(-\frac{(x-x_0)^2}{b^2}).## ##y=x-x_0 \quad x=y+x_0 \quad dy=dx.## The boundaries remain infinite, I believe. ##\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dy(y+x_0)\exp(\frac{-y^2}{b^2}).## ##\frac{2}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,y\exp(\frac{-y^2}{b^2})+\frac{2x_0}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,\exp(-\frac{y^2}{b^2}).## I then resolved the two...
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Thread 'Number of quantum accessible states of a particle given T, N?'

It's given a gas of particles all identical which has T fixed and spin S. Let's ##g(\epsilon)## the density of orbital states and ##g(\epsilon) = g_0## for ##\forall \epsilon \in [\epsilon_0, \epsilon_1]##, zero otherwise. How to compute the number of accessible quantum states of one particle? This is my attempt, and I suspect that is not good. Let S=0 and then bosons in a system. Simply, if we have the density of orbitals we have to integrate ##g(\epsilon)## and we have...
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