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 $\Delta x=L$. (a) Use the uncertainty principle to estimate the uncertainty in its momentum.(b) Because the particle stays inside the box, its average momentum must be zero. Its average squared momentum is then $\left\langle p^{2}\right\rangle=(\Delta p)^{2} .$ Estimate the energy of the particle. (c) State how the result of part (b) 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 $\Delta x=L$. (a) Use the uncertainty principle to estimate the uncertainty in its momentum.(b) Because the particle stays inside the box, its average momentum must be zero. Its average squared momentum is then $\left\langle p^{2}\right\rangle=(\Delta p)^{2} .$ Estimate the energy of the particle. (c) State how the result of part (b) compares with the actual ground-state energy.

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