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Chapter 41: Problem 47

Consider the transition from a 3\(d\) to a 2\(p\) state of hydrogen in an external magnetic field. Assume that the effects of electron spin can be ignored (which is not actually the case) so that the magnetic field interacts only with the orbital angular momentum. Identify each allowed transition by the \(m_{I}\) values of the initial and final states. For each of these allowed transitions, determine the shift of the transition energy from the zero-field value and show that there are three different transition energies.

Short Answer

Expert verified
3 different transition energies occur with shifts of \( -\mu_B B\), \( 0 \), and \(+\mu_B B\).

Step by step solution

01

Understand what the quantum numbers mean

The quantum number \(m_l\) represents the magnetic quantum number. For a \(3d\) state, the principal quantum number \(n=3\) and the azimuthal quantum number \(l=2\). So the possible \(m_l\) values are \(-2, -1, 0, 1, 2\). For a \(2p\) state, \(n=2\) and \(l=1\), with possible \(m_l\) values of \(-1, 0, +1\). Transitions between these states are allowed if \(\Delta m_l = 0, \pm1\).
02

Identify allowed transitions using \(m_l\) values

Considering the allowed transitions where \(\Delta m_l = 0, \pm 1\), we list all possible transitions. From \(3d\) \((-2, -1, 0, 1, 2)\) to \(2p\) \((-1, 0, +1)\), the allowed transitions are: \((-2 \rightarrow -1), (-1 \rightarrow 0), (0 \rightarrow -1), (0 \rightarrow 1), (1 \rightarrow 0), (1 \rightarrow 1), (2 \rightarrow 1)\).
03

Determine energy shift for each transition due to the magnetic field

The energy shift due to a magnetic field \(B\) is given by \(\Delta E = \mu_B B \Delta m_l\), where \(\mu_B\) is the Bohr magneton. For each allowed transition, calculate \(\Delta m_l\) and thus the energy shift. \((-2 \rightarrow -1): +\mu_B B\), \((-1 \rightarrow 0): +\mu_B B\), \( (0 \rightarrow -1): -\mu_B B\), \((0 \rightarrow 1): +\mu_B B\), \((1 \rightarrow 0): -\mu_B B\), \((1 \rightarrow 1): 0\), \((2 \rightarrow 1): -\mu_B B\).
04

Identify and count unique transition energies

Upon calculating the shifts, we find three unique shift values: \(-\mu_B B\), \(0\), and \(+\mu_B B\). This means that there are three distinct transition energies for these shifts, confirming the existence of three different transition energies: one shift for each value of \(\Delta E\).

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Key Concepts

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

Quantum Numbers: Decoding the Hydrogen Atom
Quantum numbers are essential in understanding the electron's position and behavior in an atom. They define the unique state of an electron. Here's a glance at what each number represents in a hydrogen atom:

- **Principal Quantum Number \( n \)**: It determines the electron's energy level and size of the orbital. The larger the number, the higher the energy level.
- **Azimuthal (Angular Momentum) Quantum Number \( l \)**: It defines the shape of the orbital and is related to the angular momentum. For a given \( n \, l \) can range from 0 to \( n-1 \).
- **Magnetic Quantum Number \( m_l \)**: This number indicates the number of orbitals and their orientation in a magnetic field. It ranges from \( -l \) to \( +l \).

For the hydrogen atom transitioning from a \(3d \) (\(n=3, l=2\)) state to a \(2p \) (\(n=2, l=1\)) state, the magnetic quantum number (\

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Most popular questions from this chapter

Show that \(\Phi(\phi)=e^{i m \phi}=\Phi(\phi+2 \pi)(\text { that is, show that }\) \(\Phi(\phi)\) is periodic with period 2\(\pi )\) if and only if \(m_{l}\) is restricted to the values \(0, \pm 1, \pm 2, \ldots . .\) (Hint: Euler's formula states that \(e^{i \phi}=\cos \phi+i \sin \phi_{\cdot} )\).

(a) Show that the total number of atomic states (including different spin states) in a shell of principal quantum number \(n\) is 2\(n^{2}\) . [Hint: The sum of the first \(N\) integers \(1+2+3+\cdots+N\) is equal to \(N(N+1) / 2 . ](\mathrm{b})\) Which shell has 50 states?

A hydrogen atom in a particular orbital angular momentum state is found to have \(j\) quantum numbers \(\frac{7}{2}\) and \(\frac{9}{2} .\) What is the letter that labels the value of \(l\) for the state?

An electron is in the hydrogen atom with \(n=5 .(a)\) Find the possible values of \(L\) and \(L_{z}\) for this electron, in units of \(\hbar\) . (b) For each value of \(L,\) find all the possible angles between \(L\) and the \(z\) -axis. (c) What are the maximum and minimum values of the magnitude of the angle between \(L\) and the \(z\) -axis?

A hydrogen atom undergoes a transition from a 2\(p\) state to the 1\(s\) ground state. In the absence of a magnetic field, the energy of the photon emitted is 122 \(\mathrm{nm}\) . The atom is then placed in a strong magnetic field in the \(z\) -direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment. (a) How many different photon wave-lengths are observed for the 2\(p \rightarrow 1\) s transition? What are the \(m_{l}\) values for the initial and final states for the transition that leads to each photon wavelength? (b) One observed wavelength is exactly the same with the magnetic field as without. What are the initial and final \(m_{l}\) values for the transition that produces a photon of this wavelength? (c) One observed wavelength with the field is longer than the wavelength without the field. What are the initial and final \(m_{l}\) values for the transition that produces a photon of this wave-length? (d) Repeat part (c) for the wavelength that is shorter than the wavelength in the absence of the field.
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