What is the value one would obtain from a single measurement of linear momentum of the particle in a box?
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Linear momentum (p) of a particle is defined as the product of its mass (m) and its velocity (v), given by the formula p = mv. Show more…
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Figure $38-13$ shows a case in which the momentum component $p_{x}$ of a particle is fixed so that $\Delta p_{x}=0 ;$ then, from Heisenberg's uncertainty principle (Eq. $38-28)$, the position $x$ of the particle is completely unknown. From the same principle it follows that the opposite is also true; that is, if the position of a particle is exactly known $(\Delta x=0)$, the uncertainty in its momentum is infinite. Consider an intermediate case, in which the position of a particle is measured, not to infinite precision, but to within a distance of $\lambda / 2 \pi$, where $\lambda$ is the particle's de Broglie wavelength. Show that the uncertainty in the (simultaneously measured) momentum component is then equal to the component itself; that is, $\Delta p_{x}=p$. Under these circumstances, would a measured momentum of zero surprise you? What about a measured momentum of $0.5 p ?$ Of $2 p ?$ Of $12 p ?$
Figure $38-13$ shows a case in which the momentum component $p_{x}$ of a particle is fixed so that $\Delta p_{x}=0 ;$ then, from Heisenberg's uncertainty principle $(E q .38-28),$ the position $x$ of the particle is completely unknown.From the same principle it follows that the opposite is also true;t hat is, if the position of a particle is exactly known $(\Delta x=0),$ the uncertainty in its momentum is infinite. Consider an intermediate case, in which the position of a particle is measured, not to infinite precision, but to within a distance of $\lambda / 2 \pi,$ where $\lambda$ is the particle's de Broglie wavelength. Show that the uncertainty in the (simultaneously measured) mo- mentum component is then equal to the component itself; that is, $\Delta p_{x}=p .$ Under these circumstances, would a measured momentum of zero surprise you? What about a measured momentum of 0.5$p ?$ Of 2$p ?$ Of 12$p ?$
(a) Find the magnitude of the momentum of a particle in a box in its nth state. (b) The minimum change in the particle's momentum that a measurement can cause corresponds to a change of ±1 in the quantum number n. If Δx = L, show that ΔpΔx ≥ ħ/2.
Adi S.
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