Find (a) the energy and (b) the magnitude of the orbital angular momentum for an electron in the \(5 d\) state of hydrogen.

The Bohr's model of the hydrogen atom is a key concept in quantum mechanics and represents a pivotal moment in the development of atomic physics. Introduced by Niels Bohr in 1913, this model proposes that an electron in an atom revolves around the nucleus in certain stable orbits without radiating energy. These orbits are known as 'stationary states' and each orbit is associated with a fixed energy level.

Energy is absorbed or emitted only when an electron transitions between these orbits, in discrete amounts known as quantum. This revolutionary idea introduced the concept of quantization to atomic structure and laid the groundwork for modern quantum mechanics. In the context of our exercise, the energy of an electron in a hydrogen atom can be calculated using the simple formula derived from Bohr's model, applying the principal quantum number corresponding to the electron's orbit.

Orbital angular momentum is a fundamental concept in quantum mechanics, analogous to the angular momentum observed in classical mechanics. For an electron in an atom, orbital angular momentum is a measure of the amount of rotation it has in its orbital path around the nucleus. However, unlike in classical mechanics, where angular momentum can take any value, in quantum mechanics, it is quantized.

This means that the electron can only possess certain discrete values of angular momentum, determined by quantum numbers. The magnitude of the orbital angular momentum is found using a formula encapsulated by the azimuthal quantum number, which defines the shape of the electron's orbit, such as s, p, d, and f orbitals. In our exercise, this concept is utilized to calculate the specific magnitude of orbital angular momentum for an electron in the '5d' state.

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. It describes phenomena that classical physics, which operates under the laws of Newtonian mechanics, cannot, such as the behavior of electrons in atoms. The theory introduces several counterintuitive concepts, including wave-particle duality, quantum entanglement, and the uncertainty principle.

In quantum mechanics, quantities that were once thought to be continuous are now understood to be quantized, existing in discrete states or levels. This is exemplified in the step-by-step solution for the hydrogen atom, where both the energy and the orbital angular momentum of the electron must adhere to specific quantized values determined by quantum numbers.

The principal quantum number, denoted by 'n', is crucial in the quantum mechanical description of an electron's state in an atom. It determines the size and energy level of the electron's orbit, or shell, around the nucleus. Values of 'n' start from 1 and increase in integer increments – the larger the value of 'n', the higher the energy level and the larger the orbit.

In the exercise, an electron is said to occupy the '5d' state, which means that the principal quantum number is 5. This indicates that the electron is in the fifth energy level, a relatively high energy state, which directly influences the energy calculation of the electron according to Bohr's model.

The azimuthal quantum number, symbolized by 'l', describes the subshell or orbital's shape within a given principal energy level and is sometimes called the 'angular momentum quantum number'. It takes on integer values ranging from 0 to (n-1), where 'n' is the principal quantum number. Different values of 'l' correspond to s, p, d, and f orbitals, representing shapes from spherical to more complex.

In our problem, the '5d' state indicates that the azimuthal quantum number 'l' is 2 (for 'd' orbitals, 'l' is always equal to 2). This number is used in the formula calculating the magnitude of the electron's orbital angular momentum, demonstrating the role of the azimuthal quantum number in determining an electron's properties.