No, a particle with a de Broglie wavelength larger than the size of its enclosed three-dimensional box cannot readily escape the boundaries of the box. This is because the de Broglie wavelength is related to the momentum of a particle, and a larger de Broglie wavelength corresponds to a smaller momentum.
When a particle is confined within a potential well, such as a three-dimensional box, its allowed energy states are quantized, forming discrete energy levels. The wave function of the particle must satisfy certain boundary conditions imposed by the box, resulting in standing wave patterns within the box.
The de Broglie wavelength of a particle is inversely proportional to its momentum. If the de Broglie wavelength is larger than the size of the box, it means the momentum of the particle is relatively small. In this case, the particle's wave function will be spread out within the box, with multiple nodes and antinodes corresponding to the standing wave pattern.
Since the particle is confined within the box, its wave function will decay rapidly outside the boundaries of the box. The probability of finding the particle outside the box is extremely low. This is due to the fact that the wave function vanishes or approaches zero at the boundaries, leading to a high probability of finding the particle within the box and a negligible probability of finding it outside.
In other words, the confinement of the particle within the box restricts its spatial extent, and the larger de Broglie wavelength does not allow it to readily escape the boundaries. The particle's behavior will be primarily localized within the box, and its escape from the box would require a significant increase in its momentum, typically through interactions or external forces.