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Chapter 5: Problem 10

A one-electron atom whose ground state is nondegenerate is placed in a uniform electric field in the \(z\)-direction. Obtain an approximate expression for the induced electric dipole moment of the ground state by considering the expectation value of ez with respect to the perturbed state vector computed to first order. Show that the same expression can also be obtained from the energy shift \(\Delta=-\alpha|\mathbf{E}|^{2} / 2\) of the ground state computed to second order. (Note: \(\alpha\) stands for the polarizability.) Ignore spin.

Short Answer

Expert verified
The induced dipole moment is proportional to the polarizability and electric field strength, linked equivalently by energy shift calculations using perturbation theory.

Step by step solution

01

Understand Perturbation Theory

In this problem, we use perturbation theory to understand the effect of an electric field on an atom. The perturbation is caused by an external electric field applied in the z-direction. The approach involves determining how the atom鈥檚 ground state is affected by this perturbation. Our primary goal is to find the induced dipole moment by evaluating the expectation value of the position operator ez for the perturbed state.
02

Write the Perturbation Hamiltonian

The perturbation Hamiltonian due to the electric field \( \mathbf{E} \) in the z-direction is given by \( H' = -eEz \). This Hamiltonian represents the potential energy change of the atom due to the external electric field in the z-direction.
03

First Order Perturbation Theory to Find Induced Dipole Moment

Use first-order perturbation theory to calculate the change in the state of the atom due to the applied perturbation. The state change is given by \( |\psi^{(1)}\rangle = \sum_{n eq 0} \frac{\langle n|H'|0 \rangle}{E_0 - E_n} |n\rangle \). The induced dipole moment \( p \) can be found using the expectation value \( \langle \psi|ez|\psi \rangle \), which simplifies to \( p = -\sum_{n eq 0} \frac{|\langle n|ez|0\rangle|^2}{E_n - E_0} \cdot eE \).
04

Obtain Energy Shift Using Second Order Perturbation Theory

The energy shift of the ground state to second order, \( \Delta E \), is given by \( \Delta E = -\sum_{n eq 0} \frac{|\langle n|H'|0 \rangle|^2}{E_n - E_0} \). Inserting \( H' = -eEz \) gives \( \Delta E = \frac{e^2E^2}{2} \sum_{n eq 0} \frac{|\langle n|z|0\rangle|^2}{E_n - E_0} \). This energy shift is expressed as \( \Delta E = -\frac{1}{2} \alpha E^2 \), where \( \alpha \) is the polarizability.
05

Equivalence of Induced Dipole Moment and Energy Shift

From Step 3, we have an expression for the induced dipole moment. From Step 4, we have the relationship \( \Delta E = -\frac{1}{2} \alpha E^2 \). The polarizability \( \alpha = 2 \sum_{n eq 0} \frac{|\langle n|z|0\rangle|^2}{E_n - E_0} \) directly links energy shifts to induced dipole moments, confirming the equivalence.

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Key Concepts

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

Induced Dipole Moment
When we talk about an induced dipole moment, we're referring to a temporary fluctuation in a molecule's electron cloud due to an external influence. In simpler terms, when an atom or molecule is placed in an external electric field, the charges within start to shift, forming a temporary dipole. This doesn't mean new charges appear. They're only rearranging to accommodate the field.

The dipole gets 'induced' because it's not a natural state of the system but a response to the electric field. In our exercise, the one-electron atom in a uniform electric field becomes polarized. The expectation value of the position operator ez helps us gauge this effect. By using perturbation theory, we find how the atom's inherent properties change under the electric field. The induced dipole moment p can simplify into an expression that shows how the shifts in charge density relate to the system's interaction with the perturbed field. Thus, the formula provides stability insights under varying external conditions.

Perturbation theory acts as a lens, translating the internal reactions of the atom when influenced by these external fields.
Polarizability
Polarizability is a measure of how easily a molecule's electron cloud can be distorted by an external electric field. It's like asking how flexible or squishy the electron cloud is when it encounters an electric force.

The greater the polarizability, the more the electron cloud can shift, indicating how susceptible the atom is to the field. It's crucial in many real-world applications, such as understanding molecular interactions and the behavior of materials in electric fields.

In our given task, the polarizability is represented by the symbol \( \alpha \) and plays a crucial role in connecting the energy shift due to the electric field to the induced dipole moment. By analyzing this, we bridge the gap between the atom's response to an external influence and its inherent ability to allow electron cloud distortion. For the one-electron atom in our exercise, knowing its polarizability helps us relate its energy shifts, calculated using second-order perturbation theory, to the induced dipole moment, confirming that \( \alpha \) is central to understanding the atom's behavior in an electric field.
Electric Field Perturbation
An electric field perturbation refers to a temporary disturbance in an atom or molecule caused by an electric field. Think of it as an external force that pushes the atom from its default state.

When we apply an electric field along a particular axis鈥攊n this case, the z-direction鈥攚e subtly change the potential energy of the atomic system. Here, the perturbation is expressed as \( H' = -eEz \), where \( e \) stands for the electron charge, \( E \) for the electric field magnitude, and \( z \) for the directional component.

This perturbation doesn't just shake things up on the surface; it forces the system to adapt, leading to changes in energy and electron density distributions. By assessing these shifts using perturbation theory, especially in first and second order, we can predict phenomena like induced dipole moments and energy shifts.

Understanding electric field perturbation is vital to interpreting how atoms or molecules behave when not at rest. With this knowledge, scientists can make accurate predictions about molecular behavior in simulations and experimental applications, enhancing their comprehension of material properties and reactions.

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Most popular questions from this chapter

Consider a particle in a two-dimensional potential $$ V_{0}= \begin{cases}0 & \text { for } 0 \leq x \leq L, 0 \leq y \leq L \\\ \infty & \text { otherwise }\end{cases} $$ Write the energy eigenfunctions for the ground and first excited states. We now add a time-independent perturbation of the form $$ V_{1}= \begin{cases}\lambda x y & \text { for } 0 \leq x \leq L, 0 \leq y \leq L \\ 0 & \text { otherwise }\end{cases} $$ Obtain the zeroth-order energy eigenfunctions and the first-order energy shifts for the ground and first excited states.

Consider a particle in one dimension moving under the influence of some timeindependent potential. The energy levels and the corresponding eigenfunctions for this problem are assumed to be known. We now subject the particle to a traveling pulse represented by a time-dependent potential, $$ V(t)=A \delta(x-c t) . $$ a. Suppose at \(t=-\infty\) the particle is known to be in the ground state whose energy eigenfunction is \(\langle x \mid i\rangle=u_{i}(x)\). Obtain the probability for finding the system in some excited state with energy eigenfunction \(\langle x \mid f\rangle=u_{f}(x)\) at \(t=+\infty\). b. Interpret your result in (a) physically by regarding the \(\delta\)-function pulse as a superposition of harmonic perturbations; recall $$ \delta(x-c t)=\frac{1}{2 \pi c} \int_{-\infty}^{\infty} d \omega e^{i \omega[(x / c)-t]} . $$ Emphasize the role played by energy conservation, which holds even quantum mechanically as long as the perturbation has been on for a very long time.

A one-dimensional harmonic oscillator is in its ground state for \(t<0\). For \(t \geq 0\) it is subjected to a time-dependent but spatially uniform force (not potential!) in the \(x\)-direction, $$ F(t)=F_{0} e^{-t / \tau} $$ a. Using time-dependent perturbation theory to first order, obtain the probability of finding the oscillator in its first excited state for \(t>0\). Show that the \(t \rightarrow \infty\) ( \(\tau\) finite) limit of your expression is independent of time. Is this reasonable or surprising? b. Can we find higher excited states?

A diatomic molecule can be modeled as a rigid rotor with moment of inertia \(I\) and an electric dipole moment \(d\) along the axis of the rotor. The rotor is constrained to rotate in a plane, and a weak uniform electric field \(\mathscr{E}\) lies in the plane. Write the classical Hamiltonian for the rotor, and find the unperturbed energy levels by quantizing the angular- momentum operator. Then treat the electric field as a perturbation, and find the first nonvanishing corrections to the energy levels.

Consider an atom made up of an electron and a singly charged \((Z=1)\) triton \(\left({ }^{3} \mathrm{H}\right)\). Initially the system is in its ground state \((n=1, l=0)\). Suppose the system undergoes beta decay, in which the nuclear charge suddenly increases by one unit (realistically by emitting an electron and an antineutrino). This means that the tritium nucleus (called a "triton") turns into a helium \((Z=2)\) nucleus of mass \(3\left({ }^{3} \mathrm{He}\right)\). a. Obtain the probability for the system to be found in the ground state of the resulting helium ion. b. The available energy in tritium beta decay is about \(18 \mathrm{keV}\) and the size of the \({ }^{3} \mathrm{He}\) atom is about \(1 \AA\). Check that the time scale \(T\) for the transformation satisfies the criterion of validity for the sudden approximation.
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