Question

A particle moving in one dimension has a state function \begin{equation} \Psi(x) = \frac{1}{(2\pi\Delta^2)^{1/4}} \exp\left(-\frac{x^2}{4\Delta^2}\right) \end{equation} where \(\Delta\) is a constant. (a) Show that the probability for the particle to have linear momentum in the range p to p + dp is P(p) dp, where \begin{equation} P(p) = \left(\frac{2}{\pi}\right)^{1/2} \left(\frac{\Delta}{\hbar}\right) \exp\left(-\frac{2p^2\Delta^2}{\hbar^2}\right). \end{equation} [Hint: Use residue theorem and the contour shown in the figure] (b) Show that the product of the uncertainties in position and momentum is \(\hbar/2\). [Hint: there is no need to calculate the standard deviations]

          A particle moving in one dimension has a state function
\begin{equation}
\Psi(x) = \frac{1}{(2\pi\Delta^2)^{1/4}} \exp\left(-\frac{x^2}{4\Delta^2}\right)
\end{equation}
where \(\Delta\) is a constant.
(a) Show that the probability for the particle to have linear momentum in the range p to p + dp is P(p) dp,
where
\begin{equation}
P(p) = \left(\frac{2}{\pi}\right)^{1/2} \left(\frac{\Delta}{\hbar}\right) \exp\left(-\frac{2p^2\Delta^2}{\hbar^2}\right).
\end{equation}
[Hint: Use residue theorem and the contour shown in
the figure]
(b) Show that the product of the uncertainties in position
and momentum is \(\hbar/2\).
[Hint: there is no need to calculate the standard
deviations]

A particle moving in one dimension has a state function

Ψ(x) = (1)/((2πΔ^2)^1/4)(-(x^2)/(4Δ^2))

where Δ is a constant.
(a) Show that the probability for the particle to have linear momentum in the range p to p + dp is P(p) dp,
where

P(p) = ((2)/(π))^1/2((Δ)/(ħ)) (-(2p^2Δ^2)/(ħ^2)).

[Hint: Use residue theorem and the contour shown in
the figure]
(b) Show that the product of the uncertainties in position
and momentum is ħ/2.
[Hint: there is no need to calculate the standard
deviations]

Added by Lu-S H.

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