Question

For a quantum particle of mass $m$ in the ground state of a square well with length $L$ and infinitely high walls, the uncertainty in position is $\Delta x \approx L$ (a) Use the uncertainty principle to estimate the uncertainty in its momentum. (b) Because the particle stays inside the box, its aver- age momentum must be zero. Its average squared momentum is then $\left\langle p^{2}\right\rangle \approx(\Delta p)^{2}$ . Estimate the energy of the particle. State how the result compares with the actual ground-state energy.

          For a quantum particle of mass $m$ in the ground state of a square well with length $L$ and infinitely high walls, the uncertainty in position is $\Delta x \approx L$ (a) Use the uncertainty principle to estimate the uncertainty in its momentum. (b) Because the particle stays inside the box, its aver- age momentum must be zero. Its average squared momentum is then $\left\langle p^{2}\right\rangle \approx(\Delta p)^{2}$ . Estimate the energy of the particle. State how the result compares with the actual ground-state energy.

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Added by Christopher L.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Keerti J

03/06/2022

Step 1

This principle can be mathematically represented as $\Delta x \Delta p \geq \frac{\hbar}{2}$, where $\Delta x$ is the uncertainty in position, $\Delta p$ is the uncertainty in momentum, and $\hbar$ is the reduced Planck's constant ($\hbar = h/2\pi$). Show more…

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For a quantum particle of mass $m$ in the ground state of a square well with length $L$ and infinitely high walls, the uncertainty in position is $\Delta x \approx L$ (a) Use the uncertainty principle to estimate the uncertainty in its momentum. (b) Because the particle stays inside the box, its aver- age momentum must be zero. Its average squared momentum is then $\left\langle p^{2}\right\rangle \approx(\Delta p)^{2}$ . Estimate the energy of the particle. State how the result compares with the actual ground-state energy.

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