Question

Selection rules in the dipole approximation are determined by the integral \(\mu_{x}^{m n}=\int \psi_{m}^{*}(\tau) \mu_{x}(\tau) \psi_{n}(\tau) d \tau .\) If this integral is nonzero, the transition will be observed in an absorption spectrum. If the integral is zero, the transition is "forbidden" in the dipole approximation. It actually occurs with low probability because the dipole approximation is not exact. Consider the particle in the one-dimensional box and set \(\mu_{x}=-e x\) a. Calculate \(\mu_{x}^{12}\) and \(\mu_{x}^{13}\) in the dipole approximation. Can you see a pattern and discern a selection rule? You may need to evaluate a few more integrals of the type \(\mu_{x}^{1 \mathrm{m}}\) The standard integral \\[ \begin{array}{c} \int x \sin \left(\frac{\pi x}{a}\right) \sin \left(\frac{n \pi x}{a}\right) d x \\ =\frac{1}{2}\left(\frac{a^{2} \cos \frac{(n-1) \pi x}{a}}{(n-1)^{2} \pi^{2}}+\frac{(n-1) \pi x}{a}\right) \\ -\frac{1}{2}\left(\frac{a^{2} \cos \frac{(n+1) \pi x}{a}}{(n+1)^{2} \pi^{2}}+\frac{a x \sin \frac{(n+1) \pi x}{a}}{(n+1) \pi}\right) \end{array} \\] is useful for solving this problem. b. Determine the ratio \(\mu_{x}^{12} / \mu_{x}^{14} .\) On the basis of your result, would you modify the selection rule that you determined in part (a)?

Short answer

The selection rule for a 1D box with a particle in a dipole approximation is that the dipole transition occurs only between states with a difference of odd integers, i.e., \(\Delta n = |m - n| = \text{odd}\). Calculating the integrals \(\mu_{x}^{12}\), \(\mu_{x}^{13}\), and \(\mu_{x}^{14}\) confirmed this pattern. The ratio \(\mu_{x}^{12} / \mu_{x}^{14}\) was indeterminate and did not provide any basis for modifying the selection rule. Therefore, the selection rule \(\Delta n = \text{odd}\) remains unchanged.

Step-by-step solution

Compute the wavefunctions for the particle in a 1D box

For a particle in a one-dimensional box of length a, the wavefunction \(\psi_n(x)\) is given by: \(\psi_n(x) = \sqrt{\frac{2}{a}} \sin{\left(\frac{n \pi x}{a}\right)}\)

Evaluate the integral for \(\mu_{x}^{12}\)

Using the given integral and wavefunctions, we can find the integral for \(\mu_{x}^{12}\): \(\mu_{x}^{12} = \int_{0}^{a} \psi_1(x) (-ex) \psi_2(x) dx\) \(\mu_{x}^{12} = -e \int_{0}^{a} \sqrt{\frac{2}{a}} \sin\left(\frac{\pi x}{a}\right) \cdot x \cdot \sqrt{\frac{2}{a}} \sin\left(\frac{2\pi x}{a}\right) dx\) Substitute the provided integral formula: \(\mu_{x}^{12} = -\frac{2e}{a}\left(\frac{a^{2} \cos \frac{\pi x}{a}}{\pi^{2}} + \frac{a x\sin \frac{3\pi x}{a}}{3\pi}\right)\Bigg|_0^a\) Evaluate the integral at the limits: \(\mu_{x}^{12} = 0\)

Evaluate the integral for \(\mu_{x}^{13}\)

Following a similar strategy, we can find the integral for \(\mu_{x}^{13}\): \(\mu_{x}^{13} = \int_{0}^{a} \psi_1(x) (-ex) \psi_3(x) dx\) \(\mu_{x}^{13} = -e \int_{0}^{a} \sqrt{\frac{2}{a}} \sin\left(\frac{\pi x}{a}\right) \cdot x \cdot \sqrt{\frac{2}{a}} \sin\left(\frac{3\pi x}{a}\right) dx\) Substitute the provided integral formula: \(\mu_{x}^{13} = -\frac{2e}{a}\left(\frac{a^{2} \cos \frac{2\pi x}{a}}{8 \pi^{2}} + \frac{a x\sin \frac{4\pi x}{a}}{4\pi}\right)\Bigg|_0^a\) Evaluate the integral at the limits: \(\mu_{x}^{13} = -\frac{2ea}{9\pi^2}\)

Determine the selection rule

From the results in Step 2 and Step 3, we can observe a pattern: \(\mu_{x}^{12} = 0\) \(\mu_{x}^{13} \neq 0\) Thus, we can infer that the selection rule for a 1D box with particle in dipole approximation is that the dipole transition occurs only between states with a difference of odd integers, i.e., \(\Delta n = |m - n| = \text{odd}\).

Evaluate the integral for \(\mu_{x}^{14}\) and find the ratio \(\mu_{x}^{12} / \mu_{x}^{14}\)

Now, we will compute the integral for \(\mu_{x}^{14}\): \(\mu_{x}^{14} = \int_{0}^{a} \psi_1(x) (-ex) \psi_4(x) dx\) \(\mu_{x}^{14} = -e \int_{0}^{a} \sqrt{\frac{2}{a}} \sin\left(\frac{\pi x}{a}\right) \cdot x \cdot \sqrt{\frac{2}{a}} \sin\left(\frac{4\pi x}{a}\right) dx\) Substitute the provided integral formula: \(\mu_{x}^{14} = -\frac{2e}{a}\left(\frac{a^{2} \cos \frac{3\pi x}{a}}{9 \pi^{2}} + \frac{a x\sin \frac{5\pi x}{a}}{5\pi}\right)\Bigg|^a_0\) Evaluate the integral at the limits: \(\mu_{x}^{14} = 0\) Calculating the ratio \(\mu_{x}^{12} / \mu_{x}^{14}\): \(\frac{\mu_{x}^{12}}{\mu_{x}^{14}} = \frac{0}{0}\) Since this ratio is indeterminate, it does not help in modifying the selection rule that we determined in part (a). Thus, the selection rule \(\Delta n = \text{odd}\) remains unchanged.

Key concepts

Dipole Approximation

In the world of quantum mechanics, the dipole approximation is a simplification used to describe how an electromagnetic field interacts with a quantum system. This approximation assumes that the wavelength of the light interacting with the system is much larger than the size of the system itself. As a result, the field is considered to be uniform across the system.

When we apply this to a simple quantum system such as an electron in an atom, the dipole approximation simplifies the complex interactions into more manageable calculations. For students and physicists, it provides an essential tool to explore quantum transitions without getting bogged down by overly complex mathematics.

In practical terms, if the dipole moment integral isn't zero, the transition can occur, leading to observable spectral lines. However, if the integral equals zero, that transition is forbidden within this approximation, although some transitions can still occur with low probability due to real-world complexities.

One-Dimensional Box

The concept of a one-dimensional box is a fundamental quantum mechanical model where a particle is restricted to move within a fixed length. This setup is a simplification to help understand the behavior of particles, like electrons, in confined spaces.

Imagine a particle trapped in a tiny box where it bounces back and forth. This system is described by a wavefunction that determines the probability of finding the particle at a particular location within the box. The wavefunctions for these systems are sine functions, with nodes at the walls of the box.

• The wavefunction for a particle in a 1D box of length \(a\) is \(\psi_n(x) = \sqrt{\frac{2}{a}} \sin{\left(\frac{n \pi x}{a}\right)}\).
• Each wavefunction corresponds to a quantized energy level defined by its quantum number \(n\).

The particle can only exist in these discrete energy levels, leading to the fascinating exploration of quantum transitions.

Quantum Transitions

Quantum transitions describe the movement of a particle between different energy states or levels within a quantum system. In the context of a one-dimensional box, these transitions occur when a particle jumps from one quantized level to another, absorbing or emitting energy.

These transitions are subject to specific rules, known as selection rules, which determine whether a transition is allowed or forbidden. For instance, in the dipole approximation, a transition is allowed if the dipole moment integral is nonzero.

• An important selection rule derived is that transitions occur between states with a change in quantum number \(\Delta n\) equal to an odd integer.
• Forbidden transitions can still happen, but with very low probability.

Quantum transitions are crucial for understanding phenomena like spectral lines in absorption and emission spectra.

Wavefunctions

Wavefunctions are fundamental to understanding the behavior of particles in quantum mechanics. They provide a mathematical description of the quantum state of a system, giving insights into properties like energy and momentum.

For a particle in a one-dimensional box, wavefunctions take a specific form that reflects the constraint of the boundaries. These wavefunctions are solutions to the Schrödinger equation, which governs the dynamics of quantum systems.

• The probability density of finding the particle is given by the square of the wavefunction's amplitude, \(|\psi(x)|^2\).
• The standing wave nature of these functions leads to nodes and antinodes within the confines of the box.

Understanding wavefunctions helps predict how particles behave under various conditions, forming the backbone of quantum mechanics and aiding in calculating properties like dipole moments in transitions.