Problem 10 - Physics for Scientists and Engineers with Modern Physics

Question

(II) An excited \(\mathrm{H}\) atom is in a 5\(d\) state. (a) Name all the states to which the atom is "allowed" to jump with the emission of a photon. (b) How many different wavelengths are there (ignoring fine structure)?

Step-by-Step Solution

Answer

Allowed transitions: 5d to 4f, 4p, 3d, 3p, 3s, 2p. Different wavelengths: 6.

Step 1: Determine the possible transitions following quantum rules

An electron in a 5d orbital can transition to orbitals having principal quantum numbers \( n < 5 \). As per selection rules for electric dipole transitions, the change in the angular momentum quantum number, \( \Delta l, \) must be \( \pm 1 \). Thus, initial \( l = 2 \) \( (d) \) can move to \( l = 1 \) (\( p \)) or \( l = 3 \) (\( f \)) orbitals. But since only levels with \( n eq 5 \) can be reached, the outer states to consider are 4f, 4p, 3d, 3p, 3s, 2p, and 2s.

Step 2: Enumerate the allowed transitions by applying quantum rules

For transitions from 5d:1. To the \( n = 4 \) level: allowed transitions are \( 4f \) and \( 4p \).2. To the \( n = 3 \) level: allowed transitions are \( 3p, \) and \( 3s \).3. To the \( n = 2 \) level: allowed transition is \( 2p \).Therefore, the allowed transitions through photon emission are from 5d to 4f, 4p, 3d, 3p, 3s, and 2p.

Step 3: Count different wavelength emissions

Each transition corresponds to a unique emission wavelength. Count the number of possible transitions determined in Step 2. Each distinct transition from the 5d level corresponds to a separate wavelength of emitted light.

Key Concepts

Selection Rules

Selection rules are guidelines used in quantum mechanics to determine the allowed transitions between different energy states of an atom or molecule. They play an essential role in predicting where an electron can move during quantum transitions, particularly when it emits or absorbs a photon.
In the context of atomic transitions, selection rules specifically dictate the changes that can occur in the quantum numbers.

• For electric dipole transitions, the change in the orbital angular momentum quantum number ( \( \Delta l \) ) must be \( \pm 1 \).
• This means that an electron in a higher energy state can only transition to a lower energy state if the change in its angular momentum is by one unit.

These rules help restrict which transitions in an atom will lead to the emission or absorption of light, making them an indispensable tool in quantum mechanics.

Angular Momentum

Angular momentum in quantum mechanics is a measure of the amount of rotation an electron has about the nucleus of an atom. There are several types of angular momentum:

• Orbital Angular Momentum: Related to the electron's motion around the nucleus, denoted by the quantum number \( l \).
• Spin Angular Momentum: Intrinsic to particles and unrelated to their motion.

For an electron in a hydrogen atom, the orbital angular momentum quantum number takes integer values from 0 up to \( n-1 \), where \( n \) is the principal quantum number. Each of these corresponds to an orbital type, like s, p, d, or f, which define the shape of the electron's path.In quantum transitions, knowing the angular momentum is vital since it influences the selection rules. Understanding these changes allows one to determine possible transitions and the specific photon emissions that result.

Electric Dipole Transitions

Electric dipole transitions are a type of transition between two energy states within an atom that occurs when a photon is either absorbed or emitted. These transitions are mainly responsible for the spectral lines observed in atomic spectra.An electric dipole transition involves the interaction between the electromagnetic field of a photon and the dipole moment of the atom. For this transition to occur:

• The change in the principal quantum number ( \( n \)) does not have specific restrictions.
• The change in the orbital angular momentum quantum number ( \( \Delta l \)) must be \( \pm 1 \), as dictated by the selection rules.

These transitions are often the most observed in atomic physics because they are highly probable, compared to other types like magnetic dipole or electric quadrupole transitions, which are less likely to occur.

Photon Emission

Photon emission is the process in which an electron transitions from a higher energy level to a lower energy level within an atom, resulting in the release of a photon. This photon carries away the energy difference between the two levels and is observed as a spectral line.Key aspects of photon emission include:

• The energy of the emitted photon is equal to the difference in energy between the two levels involved in the transition.
• The wavelength of the photon is directly proportional to this energy, following the equation \( E = h u = \frac{hc}{\lambda} \), where \( u \) is the frequency, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.

The set of all possible transitions and thus possible photon emissions in an atom helps form its unique spectra, which can be used to identify the element or determine its properties in a given environment.