One reason we don't notice quantum effects in everyday life is that Planck's constant \(h\) is so small. Treating yourself as a particle (mass \(60 \mathrm{kg}\) ) in a room-sized one-dimensional infinite square well (width \(2.6 \mathrm{m}\) ), how big would \(h\) have to be if your minimum possible energy corresponded to a speed of \(1.0 \mathrm{m} / \mathrm{s} ?\)

Planck's constant (\(h\)) is a fundamental quantity in quantum mechanics with profound implications in understanding the discrete nature of the universe. It represents the scale at which classical concepts of physics, like continuous energy, give way to quantum phenomena where energy exists in packets, or 'quanta'.

The value of Planck's constant is incredibly small, approximately equal to \(6.62607015 \times 10^{-34}\) Joule seconds. This minute number is a primary reason why quantum effects are not noticeable in our macroscopic world. However, if Planck's constant were as large as the hypothetical value suggested by the exercise, quantum behaviors would be readily observed in daily life.

The de Broglie wavelength is a concept that fuses the wave-like and particle-like properties of matter, as postulated in the wave-particle duality theory by Louis de Broglie. According to this concept, particles can exhibit wave-like characteristics, and the wavelength (\(\lambda\)) associated with a particle is inversely proportional to its momentum (\(p\)), encapsulated in the equation \(\lambda = \frac{h}{p}\).

Therefore, the idea that even large objects, like a person, have an associated wavelength is perfectly consistent within quantum mechanics, albeit typically too small to detect without extremely precise instruments.

An infinite square well is a theoretical construct used in quantum mechanics to model a particle trapped in a box with impenetrable walls. In such a scenario, the particle can bounce back and forth within the well endlessly since it possesses inadequate energy to overcome the infinitely high potential barriers.

Inside this well, a particle like an electron can only have specific energy states dictated by its quantum nature. The size of the well restricts the wavelength of the particle due to the boundary conditions: an integer multiple of half wavelengths must fit within the well. This idea leads to the quantization of energy levels and standing wave patterns within the well.

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic scales. It departs significantly from classical mechanics, which cannot adequately describe phenomena at such small scales. The principles of quantum mechanics give us insights into the peculiar behaviors of particles, where they can exist in superpositions or bound by the uncertainty principle, leading to the probabilistic nature of prediction in quantum systems.

A core concept in quantum mechanics is that particles have both wave and particle characteristics, with measurable quantities like position and momentum being fundamentally uncertain until they are observed. These concepts not only underpin our understanding of the universe but also have practical applications in technologies like semiconductors and quantum computing.