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Chapter 3: Problem 46

A spin \(\frac{3}{2}\) nucleus situated at the origin is subjected to an extemal inhomogeneous clectric field. The basic electric quadrupole interaction may by taken to be $$ H_{\mathrm{imu}}=\frac{e Q}{2 s(s-1) \hbar^{2}}\left[\left(\frac{\partial^{2} \phi}{\partial x^{2}}\right)_{0} S_{x}^{2}+\left(\frac{\partial^{2} \phi}{\partial y^{2}}\right)_{0} S_{y}^{2}+\left(\frac{\partial^{2} \phi}{\partial z^{2}}\right)_{0} S_{z}^{2}\right] $$ where \(\phi\) is the electrostatic potential satisfying Laplace's equation and the coordinate axes are so chosen that $$ \left(\frac{\partial^{2} \phi}{\partial x \partial y}\right)_{0}=\left(\frac{\partial^{2} \phi}{\partial y \partial z}\right)_{0}=\left(\frac{\partial^{2} \phi}{\partial x \partial z}\right)_{0}=0 $$ Show that the interaction energy can be written as $$ A\left(3 S_{z}^{2}-\mathrm{S}^{2}\right)+B\left(S_{+}^{2}+S_{-}^{2}\right) $$ and express \(A\) and \(B\) in terms of \(\left(\partial^{2} \phi / \partial x^{2}\right)_{0}\) and so on. Determine the energy eigenkets (in terms of \(|m\rangle\), where \(m=\pm \frac{3}{2}, \pm \frac{1}{2}\) ) and the corresponding energy eigenvalues. Is there any degeneracy?

Short Answer

Expert verified
Express coefficients as \(A = \frac{e Q}{6\hbar^{2}}\left(\frac{\partial^{2} \phi}{\partial z^{2}}\right)_0\) and \(B = \frac{e Q}{12\hbar^{2}}\left(\frac{\partial^{2} \phi}{\partial x^{2}} - \frac{\partial^{2} \phi}{\partial y^{2}}\right)_0\). The eigenstates \(|m\rangle\) are not degenerate.

Step by step solution

01

Express Hamiltonian in terms of given quantities

We start with the Hamiltonian for the system:\[H_{\mathrm{imu}} = \frac{e Q}{2 s(s-1) \hbar^{2}} \left[\left(\frac{\partial^{2}\phi}{\partial x^{2}}\right)_{0} S_{x}^{2}+\left(\frac{\partial^{2}\phi}{\partial y^{2}}\right)_{0} S_{y}^{2}+\left(\frac{\partial^{2}\phi}{\partial z^{2}}\right)_{0} S_{z}^{2}\right]\]The spin value is \(s = \frac{3}{2}\). Substituting, we can simplify our expressions later.
02

Utilize symmetry and Laplace's equation

Laplace's equation for the electrostatic potential states that:\[\left(\frac{\partial^{2} \phi}{\partial x^{2}}\right)_{0} + \left(\frac{\partial^{2} \phi}{\partial y^{2}}\right)_{0} + \left(\frac{\partial^{2} \phi}{\partial z^{2}}\right)_{0} = 0\]Thus, \(\left(\frac{\partial^{2} \phi}{\partial z^{2}}\right)_{0} = -\left(\frac{\partial^{2} \phi}{\partial x^{2}}\right)_{0} - \left(\frac{\partial^{2} \phi}{\partial y^{2}}\right)_{0}\). We will use this to express the Hamiltonian more conveniently.
03

Rewrite Hamiltonian using Spin Operators

Recall that for spin operators, \(S^{2} = S_{x}^{2} + S_{y}^{2} + S_{z}^{2}\), and that the raising and lowering operators are defined as:\[S_{\pm} = S_{x} \pm i S_{y}\]Solve for \(S_{x}^{2} + S_{y}^{2}\) using \(S^{2} - S_{z}^{2}\). This expression helps in rearranging Hamiltonian terms neatly.
04

Substitution to Express in Desired Form

Substituting \(\left(\frac{\partial^{2}\phi}{\partial z^{2}}\right)_{0}\) based on step 2, the Hamiltonian becomes:\[H_{\mathrm{imu}} = \frac{e Q}{2 \cdot \frac{3}{2}(\frac{3}{2}-1) \hbar^{2}}\left[\left(\frac{\partial^{2}\phi}{\partial x^{2}} - \frac{\partial^{2} \phi }{\partial z^{2}} \right)_0 S_x^2 - \left(\frac{\partial^{2} \phi}{\partial z^{2}} \right)_0 S_z^2\right]\]Rearranging terms with these symmetries yields the expression involving \(S_{z}^{2}\) and \(S_{\pm}\).
05

Identify Coefficients A and B

On rearranging terms using identities and symmetries, the desired form is\[H_{\mathrm{imu}} = A (3 S_{z}^{2} - S^{2}) + B (S_{+}^{2} + S_{-}^{2})\]Here, we identify\[A = \frac{e Q}{\hbar^{2}\cdot 6}\left( \frac{\partial^{2} \phi}{\partial z^{2}} \right)_{0},\B = \frac{e Q}{2\hbar^{2}\cdot 6}\left( \frac{\partial^{2} \phi}{\partial x^{2}} - \frac{\partial^{2} \phi}{\partial y^{2}} \right)_{0}\]
06

Determine Energy Eigenkets and Eigenvalues

The interaction is expressed as linear combinations of spin components. Eigenkets for \(S_{z}\) are \(|m\rangle\) where \(m = \pm \frac{3}{2}, \pm \frac{1}{2}\). The Hamiltonian perturbs these states.Energy eigenvalues include solving:\[E_{m} = A(3m^2 - \frac{15}{4}) + B ((m+1)(m+2) + (m-1)(m-2))\]To find values for each \(m\), calculate degeneracy when matrix elements are equal or zero.
07

Check for Degeneracy

Degeneracy occurs if states share the same energy. Comparing energy expressions for each of \(|m=\frac{3}{2}\rangle\), \(|m=\frac{1}{2}\rangle\), \(|m=-\frac{1}{2}\rangle\), and \(|m=-\frac{3}{2}\rangle\), solve for equal energy values. Note symmetry or parameter configurations result in equal energies for different states.

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Key Concepts

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

Spin Quantum Number
In quantum mechanics, the concept of spin plays a crucial role in understanding the intrinsic angular momentum of particles. The spin quantum number, often denoted as \(s\), characterizes the intrinsic angular momentum attributed to elementary particles. For a spin-\(\frac{3}{2}\) system, as used in our exercise, the total spin \(S\) can take values derived from multiplying \(s\) with its quantum numbers.

Spin has both magnitude and direction, which are described using quantum mechanics. The possible projections of spin along a specific axis, typically the \(z\)-axis, are referred to as \(m_s\) quantized values ranging from \(-s\) to \(+s\). In our scenario, for \(s=\frac{3}{2}\), the allowed values of \(m_s\) are \(\pm\frac{3}{2}, \pm\frac{1}{2}\). These values give rise to the eigenstates \(|m\rangle\).

The spin quantum number is a fundamental attribute, similar to mass or charge, affecting how particles interact with fields, contributing to the total representation of a state.
Electric Quadrupole Interaction
Electric quadrupole interaction pertains to the coupling between the electric field produced by a charge distribution and the electric quadrupole moment of a nucleus. This interaction is especially significant for nuclei with spin greater than \(\frac{1}{2}\), such as our spin-\(\frac{3}{2}\) system. A nucleus with a quadrupole moment interacts with electric field gradients due to its non-spherical charge distribution.

In the exercise, the Hamiltonian expresses this interaction using the electrostatic potential \(\phi\), indicating how changes or gradients in the potential affect the energy levels. The Hamiltonian's complexity involves partial second derivatives of \(\phi\), emphasizing the higher order terms in a Taylor expansion that describe quadrupole interactions.

The electric quadrupole interaction thereby introduces energy shifts which play a role in fine structure splitting observed in atomic spectra and can cause significant effects in nuclear magnetic resonance experiments. Its inclusion is vital for describing systems where the electric and magnetic properties of the nucleus influence the system's overall energy.
Energy Eigenvalues
Energy eigenvalues in quantum mechanics signify the possible outcomes of measuring the energy of a system. For a quantized system, as presented in our exercise, these energies result from solving the Schr枚dinger equation for the Hamiltonian. Each eigenvalue corresponds to a specific eigenstate or eigenket \(|m\rangle\).

In our scenario, energy eigenvalues are affected by both parameters \(A\) and \(B\), which involve derivatives of the electrostatic potential. These eigenvalues are calculated using the Hamiltonian's expression associated with respective \(|m\rangle\) states. Mathematically, for each \(m\), the energy eigenvalue \(E_{m}\) is influenced by contributions from both \(3S^2_z - S^2\) and \(S_{+}^2 + S_{-}^2\) operators.

Evaluating these values allows us to understand which energy states are accessible under the influence of an inhomogeneous electric field. Notably, finding equal values amongst different \(m\) states can reveal degeneracy, where multiple quantum states share identical energy levels.
Hamiltonian
The Hamiltonian of a system is a cornerstone concept in quantum mechanics, representing the total energy operator of that system. In our exercise, the Hamiltonian \(H_{\mathrm{imu}}\) for a spin-\(\frac{3}{2}\) nucleus under an inhomogeneous electric field is given by the electric quadrupole term. It incorporates the components of the quantum spin operators \(S_x, S_y, S_z\) in relation to changes in the electrostatic potential \(\phi\).

The exercise simplifies interaction terms into the form \(A(3S_z^2 - S^2) + B(S_+^2 + S_-^2)\), distinguishing between contributions from spin components along and perpendicular to the field gradient. This decomposition highlights symmetry properties and offers a clearer view of the physical situation.

The Hamiltonian is crucial as it dictates the system's dynamics, described by its eigenvalues and eigenkets. Its formulation from physical conditions provides insights into how a system might respond under various external influences, like electric and magnetic fields, and is essential for predicting experimental outcomes. The precision in expressing the Hamiltonian helps anticipate phenomena such as energy shifts and degeneracies in quantum systems.

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Most popular questions from this chapter

Let the Hamiltonian of a rigid body be $$ H=\frac{1}{2}\left(\frac{K_{1}^{2}}{I_{1}}+\frac{K_{2}^{2}}{I_{2}}+\frac{K_{3}^{2}}{I_{3}}\right) $$ where \(\mathbf{K}\) is the angular momentum in the body frame. From this expression obtain the Heisenberg equation of motion for \(\mathbf{K}\) and then find Euler's equation of motion in the correspondence limit.

An angular-momentum cigenstate \(\left\lfloor j, m=m_{\max }=j\right\rangle\) is rotated by an infinitesimal angle \(\varepsilon\) about the \(y\)-axis. Without using the explicit form of the \(d_{m^{\prime} m}^{(f)}\) function, obtain an expression for the probability for the new rotated state to be found in the original state up to terms of order \(\varepsilon^{2}\).

Consider the Coulomb potential \(V(\mathbf{x})=-Z e^{2} / r\) and define the (quantum-mechanical operator analogue of the) Runge-Lenz vector $$ \mathbf{M}=\frac{1}{2 m}(\mathbf{p} \times \mathbf{L}-\mathbf{L} \times \mathbf{p})-\frac{Z e^{2}}{r} \mathbf{x} $$ Prove that \(\mathbf{M}\) is Hermitian and that it commutes with the Hamiltonian. We will retum to \(\mathbf{M}\) when we go through Pauli's algebraic solution for this Hamiltonian in Section 4.1.4.

A particle in a spherically symmetrical potential is known to be in an eigenstate of \(\mathbf{L}^{2}\) and \(L_{2}\) with eigenvalues \(\hbar^{2} l(l+1)\) and \(m \hbar\), respectively. Prove that the expectation values between \(|I m\rangle\) states satisfy $$ \left\langle L_{x}\right\rangle=\left\langle L_{y}\right\rangle=0, \quad\left\langle L_{x}^{2}\right\rangle=\left\langle L_{y}^{2}\right\rangle=\frac{\left[l(l+1) \hbar^{2}-m^{2} \hbar^{2}\right]}{2} . $$ Interpret this result semiclassically.

a. Using the fact that \(J_{x}, J_{y}, J_{z}\), and \(J_{\pm} \equiv J_{x} \pm i J_{y}\) satisfy the usual angular-momentum commutation relations, prove that $$ \mathrm{J}^{2}=J_{x}^{2}+J_{+} J_{-}-\hbar J_{z^{+}} $$b. Using this result, or otherwise, derive the coefficient \(c\) - that appears in $$ J_{-}|j m\rangle=c_{-}|j, m-1\rangle . $$
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