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I have a problem finding ##\left|Ψ(x,t)\right|^2## from the following equation:

$$Ψ(x,t) = \frac 1 {\pi \sqrt{2a}} \int_{-∞}^{+∞} \frac {\sin(ka)} k e^{i(kx - \frac {ħk^2} {2m} t)} dk$$

and tried to plot like the pic below (Source Introduction to quantum mechanics by David. J. Griffiths, 2nd Edition, page 75). I need help finding ##\left|Ψ(x,t)\right|^2##.

Does understanding Plancherel's theorem helps understanding the expression of wave packet as follows?

$$Ψ(x,t) = \frac 1 {\sqrt{2\pi}} \int_{-∞}^{+∞} \phi(k) e^{i(kx - \frac {ħk^2} {2m} t)} dk$$

I didn't understand how Griffiths arrived at that expression. Thanks in advance.

$$Ψ(x,t) = \frac 1 {\pi \sqrt{2a}} \int_{-∞}^{+∞} \frac {\sin(ka)} k e^{i(kx - \frac {ħk^2} {2m} t)} dk$$

and tried to plot like the pic below (Source Introduction to quantum mechanics by David. J. Griffiths, 2nd Edition, page 75). I need help finding ##\left|Ψ(x,t)\right|^2##.

Does understanding Plancherel's theorem helps understanding the expression of wave packet as follows?

$$Ψ(x,t) = \frac 1 {\sqrt{2\pi}} \int_{-∞}^{+∞} \phi(k) e^{i(kx - \frac {ħk^2} {2m} t)} dk$$

I didn't understand how Griffiths arrived at that expression. Thanks in advance.