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The probability density of finding a classical particle between x and $x+\text{\Delta}x$ depends on how much time $\text{\Delta}t$ the particle spends in this region. Assuming that its speed u is constant, this time is $\text{\Delta}t=\text{\Delta}x\text{/}u,$ which is also constant for any location between the walls. Therefore, the probability density of finding the classical particle at x is uniform throughout the box, and there is no preferable location for finding a classical particle. This classical picture is matched in the limit of large quantum numbers. For example, when a quantum particle is in a highly excited state, shown in [link] , the probability density is characterized by rapid fluctuations and then the probability of finding the quantum particle in the interval $\text{\Delta}x$ does not depend on where this interval is located between the walls.
The ground state of the cart, treated as a quantum particle, is
Therefore, $n={(K\text{/}{E}_{1})}^{1\text{/}2}={(0.050\text{/}1.700\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-68}})}^{1\text{/}2}=1.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{33}$ .
Check Your Understanding (a) Consider an infinite square well with wall boundaries $x=0$ and $x=L$ . What is the probability of finding a quantum particle in its ground state somewhere between $x=0$ and $x=L\text{/}4$ ? (b) Repeat question (a) for a classical particle.
a. 9.1%; b. 25%
Having found the stationary states ${\psi}_{n}(x)$ and the energies ${E}_{n}$ by solving the time-independent Schrӧdinger equation [link] , we use [link] to write wave functions ${\text{\Psi}}_{n}(x,t)$ that are solutions of the time-dependent Schrӧdinger’s equation given by [link] . For a particle in a box this gives
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