In baseball, a \(100 .-\mathrm{g}\) ball can travel as fast as \(100 . \mathrm{mph}\). What is the de Broglie wavelength of this ball? The Voyager spacecraft, with a mass of about \(250 . \mathrm{kg}\), is currently travelling at \(125,000 \mathrm{~km} / \mathrm{h}\). What is its de Broglie wavelength?

Quantum mechanics is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles. Traditional physics, known as classical mechanics, could not explain phenomena at the atomic and subatomic level. Quantum mechanics provides a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It underpins a host of modern technologies, such as the laser and semiconductor devices that are vital to modern computers and smartphones.

In quantum mechanics, particles have properties of both particles and waves, and one can only predict the probability of where a particle might be – which is fundamentally different from classical mechanics, where objects can have determined positions and velocities. The equation that connects classical and quantum physics is the de Broglie wavelength formula, which implies that matter can exhibit wave-like behavior under certain conditions.

Planck's constant, symbolized as \( h \), is a fundamental constant that is very important in the field of quantum mechanics. Its value is approximately \( 6.626 \times 10^{-34} \) joule-seconds. This constant is crucial in the aspect of quantization of energy and has dimensions of energy multiplied by time, which are units of action.

When Max Planck was studying blackbody radiation, he proposed that energy is quantized, and can be given as an integer multiple of an elementary unit \( E = h u \), where \( u \) is the frequency. This discovery led to the quantum revolution in physics. Planck’s constant makes it possible to describe the particle aspect of light and other quantum particles. It is central to the theory of quantum mechanics, appearing in many fundamental equations, including the one used to find the de Broglie wavelength of a particle, highlighting the wave-particle duality.

Wave-particle duality is a fundamental concept of quantum mechanics which states that elementary particles such as electrons and photons exhibit both wave-like and particle-like properties. French physicist Louis de Broglie first introduced wave-particle duality in his PhD thesis in 1924, suggesting that just as light has both wave-like and particle-like properties, matter also should have wave-like properties.

The de Broglie wavelength is the wavelength, \( \lambda \), associated with a particle and is related to its momentum, \( p \), through the Planck constant, \( h \): \[ \lambda = \frac{h}{p} \]. The de Broglie hypothesis was confirmed by electron diffraction experiments. For macroscopic objects, such as a baseball or a spacecraft, the associated de Broglie wavelength is usually extremely small, making their wave-like properties very difficult to detect and practically negligible in the classical perspective.