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Chapter 31: Problem 45

Consider two different states of a hydrogen atom. In state I the maximum value of the magnetic quantum number is \(m_{\ell}=3 ;\) in state II the corresponding maximum value is \(m_{\ell}=2 .\) Let \(L_{1}\) and \(L_{\mathrm{II}}\) represent the magnitudes of the orbital angular momentum of an electron in states I and II, respectively. (a) Is \(L_{1}\) greater than, less than, or equal to \(L_{\mathrm{II}}\) ? Explain. (b) Calculate the ratio \(L_{\mathrm{I}} / L_{\mathrm{II}}\).

Short Answer

Expert verified
(a) \( L_1 > L_{II} \) because \( \sqrt{12} > \sqrt{6} \). (b) Ratio is \( \sqrt{2} \).

Step by step solution

01

Determine the Quantum Number for State I

In state I, the maximum magnetic quantum number \(m_\ell \) is given as 3. The azimuthal quantum number \( \ell \) that can have this maximum value also equals 3. Therefore, \( \ell_1 = 3 \).
02

Determine the Quantum Number for State II

For state II, the maximum magnetic quantum number \(m_\ell \) is 2. Thus, the azimuthal quantum number \( \ell \) in this case also equals 2. Hence, \( \ell_{II} = 2 \).
03

Calculate Orbital Angular Momentum for State I

The magnitude of the orbital angular momentum \(L\) for any state is given by the formula \[ L = \sqrt{ \ell(\ell+1) } \hbar \]\where \(\hbar\) is the reduced Planck's constant. Thus, for state I, \[ L_I = \sqrt{3(3+1)} \hbar = \sqrt{12} \hbar \]
04

Calculate Orbital Angular Momentum for State II

Similarly, for state II, the magnitude of the orbital angular momentum is given by \[ L_{II} = \sqrt{2(2+1)} \hbar = \sqrt{6} \hbar \]
05

Compare Magnitudes of Orbital Angular Momentum

Compare \( L_I = \sqrt{12} \hbar \) and \( L_{II} = \sqrt{6} \hbar \). Since \( \sqrt{12} > \sqrt{6} \, L_I > L_{II} \).
06

Calculate the Ratio of Angular Momenta

The ratio of the angular momentum magnitudes is given by \[ \frac{L_I}{L_{II}} = \frac{\sqrt{12}\hbar}{\sqrt{6}\hbar} = \frac{\sqrt{12}}{\sqrt{6}} = \sqrt{2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hydrogen Atom
The hydrogen atom is the simplest atom in the universe. It consists of one proton and one electron. The proton acts as the nucleus and the electron orbits around it. The study of the hydrogen atom is fundamental in quantum mechanics because it illustrates basic quantum concepts.
  • The electron can only exist in discrete energy levels, called orbitals. These orbitals are defined by specific quantum numbers.
  • Each electron orbit or shell is represented by a principal quantum number, denoted as \(n\).
The hydrogen atom is important in physics because it helps scientists understand the behavior of atoms and the structure of the universe. By observing and predicting the energy levels and spectral lines of hydrogen, scientists can develop theories about light, matter, and atomic interactions.
Magnetic Quantum Number
The magnetic quantum number, denoted as \(m_\ell\), is one of three quantum numbers that describe orbitals in an atom.
  • It specifies the orientation of the orbital angular momentum vector in space.
  • For a given azimuthal (or angular momentum) quantum number (\(\ell\)), the magnetic quantum number can take integer values from \(-\ell\) to \(+\ell\).
  • This means if \(\ell = 3\), then \(m_\ell\) can be -3, -2, -1, 0, +1, +2, or +3.
The magnetic quantum number is crucial as it determines the number of orbitals and their orientation in a magnetic field. For a hydrogen atom, these properties influence how it interacts with magnetic fields and how spectral lines are split鈥攁 phenomenon known as the Zeeman effect.
Orbital Angular Momentum
Orbital angular momentum refers to the angular momentum of an electron in an atom due to its motion around the nucleus.
  • The magnitude of the orbital angular momentum is described by the formula: \[ L = \sqrt{\ell (\ell + 1)} \hbar \]where \(\hbar\) is the reduced Planck's constant and \(\ell\) is the azimuthal quantum number.
  • This formula shows that as the value of \(\ell\) increases, the magnitude of the angular momentum also increases.
  • Orbital angular momentum is quantized, meaning it can only take specific values determined by \(\ell\).
Understanding orbital angular momentum helps explain the behavior of electrons in atoms and the rules governing transitions between different atomic states. These transitions are responsible for the absorption and emission spectra of atoms.

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Most popular questions from this chapter

Rydberg Atoms There is no limit to the size a hydrogen atom can attain, provided it is free from disruptive outside influences. In fact, radio astronomers have detected radiation from large, so-called "Rydberg atoms" in the diffuse hydrogen gas of interstellar space. (a) Find the smallest value of \(n\) such that the Bohr radius of a single hydrogen atom is greater than 8.0 microns, the size of a typical single-celled organism. (b) Find the wavelength of radiation this atom emits when its electron drops from level \(n\) to level \(n-1\). (c) If the electron drops one more level, from \(n-1\) to \(n-2,\) is the emitted wavelength greater than or less than the value found in part (b)? Explain.

The electronic configuration of a given atom is \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{1} .\) How many electrons are in this atom?

Consider the following four transitions in a hydrogen atom: (i) \(n_{i}=2, n_{f}=6\) (ii) \(n_{i}=2, n_{\mathrm{f}}=8\) (iii) \(n_{i}=7, n_{f}=8\) (iv) \(n_{i}=6, n_{f}=2\) Find (a) the longest-and (b) the shortest-wavelength photon that can be emitted or absorbed by these transitions. Give the value of the wavelength in each case. (c) For which of these transitions does the atom lose energy? Explain.

(a) In an \(X\) -ray tube, do you expect the wavelength of the characteristic X-rays to increase, decrease, or stay the same if the energy of the electrons striking the target is in creased? (b) Choose the best explanation from among the following: I. Increasing the energy of the incoming electrons will in crease the wavelength of the emitted \(X\) rays. II. When the energy of the incoming electrons is increased, the energy of the \(X\) -rays is also increased; this, in turn, decreases the wavelength. III. The wavelength of characteristic X-rays depends only on the material used in the metal target, and does not change if the energy of incoming electrons is increased.

Photorefractive Keratectomy A person's vision may be improved significantly by having the cornea reshaped with a laser beam, in a procedure known as photorefractive keratectomy. The excimer laser used in these treatments produces ultraviolet light with a wavelength of \(193 \mathrm{nm}\). (a) What is the difference in energy between the two levels that participate in stimulated emission in the excimer laser? (b) How many photons from this laser are required to deliver an energy of \(1.58 \times 10^{-13} \mathrm{J}\) to the cornea?
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