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The probability density of finding a classical particle between x and depends on how much time the particle spends in this region. Assuming that its speed u is constant, this time is 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 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, .
Check Your Understanding (a) Consider an infinite square well with wall boundaries and . What is the probability of finding a quantum particle in its ground state somewhere between and ? (b) Repeat question (a) for a classical particle.
a. 9.1%; b. 25%
Having found the stationary states and the energies by solving the time-independent Schrӧdinger equation [link] , we use [link] to write wave functions 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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