Question

Show that a transition where \(\mathrm{Am},=\pm 1\) conesponds to a dipole moment in the \(x y\) -plane. while \(1 m_{t}=0\) corresponds to a momentalong the z-avis. (You need consider only the \(\phi\) -parts of \(\psi_{1}\) and \(\psi_{\rho}\) which are of the form \(e^{-m}\).)

Short answer

A transition where \(m=\pm1\) corresponds to a dipole moment in the \(x y\) -plane, while \(m_{t}=0\) corresponds to a moment along the \(z\)-axis. This conclusion is based on the calculations of the dipole moment components in spherical coordinates and their dependence on the values of \(m\) or \(m_{t}\).

Step-by-step solution

Identify wavefunctions and express in terms of Euler's formula

Start with the \(\phi\)-parts of the wavefunctions \(\psi_{1}\) and \(\psi_{\rho}\) which are \(e^{-im\phi}\) and \(e^{-im_{t}\phi}\) respectively. Using Euler’s formula, express these terms as \(cos(m\phi) - i sin(m\phi)\) and \(cos(m_{t}\phi) - i sin(m_{t}\phi)\) respectively.

Calculation of dipole moment components

The dipole moment vector components in spherical coordinates are: \(d_x = r sin(\theta) cos(\phi)\), \(d_y = r sin(\theta) sin(\phi)\), and \(d_z = r cos(\theta)\). Calculate the \(x y\)-plane components of dipole moment for \(m=\pm1\) and \(m_{t}=0\) using the wavefunctions. For \(m=\pm1\), \(d_x = r sin(\theta) (cos(\phi) + sin(\phi))\) and \(d_y = r sin(\theta) (cos(\phi) - sin(\phi))\). For \(m_{t}=0\) , \(d_z = r cos(\theta)\).

Discussion of results

When \(m=\pm1\), the dipole moment components in the \(x y\)- plane are present. This means that transitions where \(m=\pm1\) corresponds to a dipole moment in the \(x y\)-plane. For \(m_{t}=0\), the \(z\)-component of the dipole moment is present while other components vanish, which indicates that it corresponds to a moment along the \(z\)-axis.

Key concepts

Wavefunctions in Quantum Mechanics

At the heart of quantum mechanics lie wavefunctions, which are mathematical descriptions of the quantum state of a system. The wavefunction is denoted by the symbol \( \psi \) and embodies all information about a system.

When solving quantum mechanical problems, such as analyzing the behavior of particles in an atom, we represent these particles by their wavefunctions. In the context of dipole moments, these wavefunctions can be used to determine the distribution of charge and, consequently, the electric dipole moment.

To comprehend the results of quantum measurements, physicists often work with probabilities, since wavefunctions give the probability amplitude, which when squared (\( |\psi|^2 \)), provides the probability density of finding a particle at a particular location in space.

Euler's Formula in Physics

Euler's formula, a gem in the crown of mathematical physics, plays a pivotal role in linking algebra to trigonometry and complex numbers. It states that for any real number \( \phi \), \( e^{i\phi} = \cos(\phi) + i \sin(\phi) \).

In physics, this formula comes in handy when dealing with oscillations and wave-like phenomena, which appear frequently in quantum mechanics. For example, the \( \phi \) part of the wavefunctions in our exercise can be expressed as complex exponentials using Euler's formula, which simplifies the calculation of dipole moments in quantum systems. By converting complex exponentials into sines and cosines, the physical interpretation of these wavefunctions, particularly in terms of their phase relationships, becomes more tangible.

Spherical Coordinates in Dipole Moment

Spherical coordinates are a three-dimensional coordinate system where any point in space is described by three values: the radial distance (\(r\)), the polar angle (\(\theta\)), and the azimuthal angle (\(\phi\)). This system is especially useful in physics when dealing with problems having spherical symmetry, such as those involving atoms and molecules.

In the calculation of dipole moments for systems with spherical symmetry, using spherical coordinates allows us to express the dipole moment vector components intuitively as a function of these angles. The exercise shows how one can compute these components for different quantum states denoted by quantum numbers related to angular momentum, simplifying the determination of the orientation of the dipole moment.

Dipole Transition in Quantum Physics

Dipole transitions are quantum jumps between energy levels in an atom or molecule that involve the absorption or emission of a photon and are governed by specific selection rules. These transitions result in changes in the electric dipole moment, a vector quantity that represents the separation of positive and negative charges within the atom or molecule.

The exercise demonstrates how transitions between specific quantum states—characterized by their quantum numbers—correlate with the orientation of the dipole moment within the atomic or molecular system. In particular, the correlation between the quantum number \(m\) and the dipole moment in either the \(xy\)-plane or along the \(z\)-axis is crucial to understanding how absorbed or emitted radiation affects the movement of charged particles.