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A homework question that I don't conceptually understand:

A quantum particle of mass M is trapped inside an infinite, one-dimensional square well of width $L$. If we were to solve Schrodinger's wave equation for such a particle, we could show that the minimum energy such a particle can have is

$$E_{min} = \frac{h^2}{8ML^2}$$

where $E$ is measured from the bottom of the well and $h$ is Planck's constant. Estimate the minimum energy of the same particle using the Heisenberg Uncertainty Principle and assess the consistency of your estimate with the exact result.

What I don't understand is how can I equate the HUP to this since I don't know what the particle's momentum is in relation to it's uncertainty. Thanks for any clarification.

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Heisenberg's uncertainty principle is

$$\Delta x \Delta p \geq \hbar/2.$$

Since the well is of width $L$, you have a measure for the uncertainty on the position $\Delta x$. Then assume the lowest possible value for $\Delta p$, i.e. the one for which the above inequality becomes an equality. Lastly, use $E = \dfrac{p^2}{2m}$ to find an expression for $E$.

A useful question to look at as well might be this one.

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  • $\begingroup$ If I know L, doesn't that make the uncertainty of Δx 0, which would mean the uncertainty for Δp is infinite? $\endgroup$
    – user42079
    Mar 8, 2014 at 1:23

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