Using the de Broglie wavelength, \(h / p,\) show that orbits whose angular momentum is quantized according to the Bohr quantization condition \((J=n h / 2 \pi)\) correspond to orbits whose circumference is an integer number of wavelengths.

The de Broglie wavelength is a fundamental concept in quantum mechanics. It describes the wave-like properties of particles. According to de Broglie, every particle has a wavelength given by \( \lambda = \frac{h}{p} \). Here, \( h \) is the Planck constant and \( p \) is the momentum of the particle. This means that even particles such as electrons exhibit wave-like behavior. In essence, the de Broglie wavelength bridges the gap between particle and wave theories, showing that particles can behave like waves under certain conditions.

Angular momentum is a measure of the extent of rotation an object has, taking into account its mass, shape, and rotational velocity. In quantum mechanics, the angular momentum \( J \) of an electron in orbit is quantized. The Bohr quantization condition gives us the quantized angular momentum as: \[ J = n \frac{h}{2\pi} \] Here, \( n \) is an integer known as the principal quantum number, and \( h \) is again the Planck constant. More importantly, for a particle moving in a circular orbit of radius \( r \), the angular momentum is related to the particle’s momentum \( p \) as: \[ J = rp \] By combining these relationships, we can connect the small-scale world of quantum mechanics with macroscopic observations.

The principal quantum number, denoted \(n\), is a key component in quantum mechanics. It describes the energy level of an electron in an atom. This integer value starts from \(n = 1 \) and increases. The higher the principal quantum number, the higher the energy level and the farther the electron is from the nucleus. The principal quantum number also determines other properties such as angular momentum. For example, in Bohr's model of the hydrogen atom, the quantization condition for angular momentum is given by: \[ J = n \frac{h}{2\pi} \] This demonstrates that only specific orbits with defined circumference (related to de Broglie wavelength) are allowed, linking the particle’s position to its quantum state.

The Planck constant, denoted as \(h\), is a fundamental constant in physics. It connects the energy of a photon to its frequency in the equation \[ E = h u \] where \( E \) is energy and \( u \) is the frequency. The Planck constant has a very small value, \( 6.626 \times 10^{-34} \) Joule seconds (J·s). In the context of the de Broglie wavelength and Bohr quantization condition, \( h \) serves as a crucial link between a particle’s wave and its momentum. For instance, the de Broglie wavelength is given by \[\lambda = \frac{h}{p} \] and the angular momentum of an electron is quantized according to \[ J = n \frac{h}{2\pi} \]. These relationships show how the Planck constant acts as a bridge between the energy levels of electrons in quantum states and their corresponding wave-like properties.