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Chapter 5: Problem 21

Suppose the electron had a very small intrinsic electric dipole moment analogous to the spin magnetic moment (that is, \(\mu_{e l}\) proportional to \(\sigma\) ). Treating the hypothetical \(-\mu_{e l} \cdot \mathbf{E}\) interaction as a small perturbation, discuss qualitatively how the energy levels of the Na atom \((Z=11)\) would be altered in the absence of any external electromagnetic field. Are the level shifts first order or second order? State explicitly which states get mixed with each other. Obtain an expression for the energy shift of the lowest level that is affected by the perturbation. Assume throughout that only the valence electron is subjected to the hypothetical interaction.

Short Answer

Expert verified
The hypothetical interaction causes first-order energy shifts in the Na atom's valence electron levels. States with the same \(l, m\) but different spin mix. Energy shift for the 3s state is \(-\mu_{el} \cdot \langle \mathbf{E}_\text{eff} \rangle\).

Step by step solution

01

Identify the Problem

We have to consider a hypothetical intrinsic electric dipole moment of an electron, similar to its spin magnetic moment, and analyze how this perturbation affects the energy levels of a sodium (Na) atom, specifically considering first-order perturbation theory and state mixing.
02

Understand the Perturbation Term

The perturbation term is given by \(-\mu_{el} \cdot \mathbf{E}\), where \(\mu_{el}\) is analogous to the electron's spin magnetic moment and \(\mathbf{E}\) is the electric field. Since there is no external field, the interaction is internal to the atom.
03

Apply Perturbation Theory

In perturbation theory, the first-order energy shift \(\Delta E^{(1)}\) of a state \(|\psi_n\rangle\) due to a perturbation \(\hat{H}'\) is given by \(\Delta E^{(1)} = \langle \psi_n| \hat{H}' | \psi_n \rangle\). Here, the perturbation is \(-\mu_{el} \cdot \mathbf{E}\), and we consider this as a small perturbation.
04

Consider State Mixing

States with the same angular momentum quantum numbers \(l, m\) but different spins \(s\) may be mixed. In the Na atom, this means mixing between different states of the valence electron that have the same principal quantum number \(n\) but differ in their spin orientations.
05

Analyze the Energy Shift Order

Since we are considering a first-order perturbation, the energy shifts are first order. However, in the absence of an external electromagnetic field, any mixing or shift would be dependent on internal fields, and only the spin states of electrons could be affected initially.
06

Determine the Lowest Affected Level

The lowest energy level in the Na atom that can be affected is the ground state of the valence electron, which is in the \(3s\) orbital. Its shift would primarily depend on the internal electric field it experiences, considering the electron's hypothetical electric dipole moment.
07

Expression for Energy Shift

The expression for the energy shift of the lowest level due to the perturbation can be given by\[\Delta E = -\mu_{el} \cdot \langle \mathbf{E}_\text{eff} \rangle\]where \(\langle \mathbf{E}_\text{eff} \rangle\) is the effective internal electric field felt by the valence electron in the absence of external fields.

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Key Concepts

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

Intrinsic Electric Dipole Moment
An intrinsic electric dipole moment refers to the idea that a particle, such as an electron, might possess a small but inherent separation of electrical charges within itself. This concept is akin to having an internal electric field. In our hypothetical scenario, we imagine the electron as having an electric dipole moment much like its spin magnetic moment. The challenge is to understand how this impacts the behavior of the electron when no external fields are present.
  • This intrinsic property can interact with other internal fields within an atom, causing shifts in the energy levels of the atom's electrons.
  • The effect in quantum mechanics is considered as a small perturbation, which can slightly alter the system's properties.
Even though real electrons do not have a measureable intrinsic electric dipole moment based on current scientific consensus, exploring this hypothetical situation helps deepen our understanding of atomic behavior and influences in quantum mechanics.
Sodium Atom Energy Levels
The sodium atom, with an atomic number of 11, has a single valence electron in its outermost shell. This electron primarily resides in the 3s orbital of the ground state when unperturbed. Examining the energy levels:
  • Given quantum mechanical principles, each electron in an atom occupies specific energy levels or orbitals.
  • The energy levels are partly determined by the electrons鈥� interactions with one another and the nucleus.
  • In sodium, the valence electron's energy state can be slightly shifted by various internal perturbations.
When subjected to a hypothetical intrinsic electric dipole moment, the energy levels of sodium might experience minor shifts. This adjustment would be first-order based on standard quantum perturbation theory. In essence, these perturbations are small separations from the usual state adduced by internal interactivity akin to an effective electric field generated within the atomic structure.
State Mixing
State mixing in quantum mechanics refers to a situation where different quantum states with similar energies can combine or "mix" with each other. This happens when a perturbation causes these states to interfere and share qualities of one another. When discussing state mixing in a sodium atom:
  • Primarily involves the valence electron's states that have the same principal quantum number but different spin orientations.
  • The intrinsic electric dipole presence pushes different spin states to interact, potentially leading to a combination or mixing of these states.
  • This mixing is a significant phenomenon influencing how energy levels are shifted under perturbation theory.
The effect is more pronounced in systems with no external electromagnetic fields due to their reliance on internal attributes. This reveals deeper interactions between quantum states and their dependency on minor perturbations, offering a refined understanding of atomic systems.

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Most popular questions from this chapter

Evaluate the matrix elements (or expectation values) given below. If any vanishes, explain why it vanishes using simple symmetry (or other) arguments. a. \(\langle n=2, l=1, m=0|x| n=2, l=0, m=0\rangle\). b. \(\left\langle n=2, l=1, m=0\left|p_{z}\right| n=2, l=0, m=0\right\rangle\). [In (a) and (b), \(|n l m\rangle\) stands for the energy eigenket of a nonrelativistic hydrogen atom with spin ignored.] c. \(\left\langle L_{z}\right\rangle\) for an electron in a central field with \(j=\frac{9}{2}, m=\frac{7}{2}, l=4\). d. \(\left\langle\right.\) singlet, \(m_{s}=0\left|S_{z}^{(e-)}-S_{z}^{(e+)}\right|\) triplet, \(\left.m_{s}=0\right\rangle\) for an \(s\)-state positronium. e. \(\left\langle\mathbf{S}^{(1)} \cdot \mathbf{S}^{(2)}\right\rangle\) for the ground state of a hydrogen molecule.

Compute the Stark effect for the \(2 s_{1 / 2}\) and \(2 p_{1 / 2}\) levels of hydrogen for a field \(\mathscr{E}\) sufficiently weak so that \(e \mathscr{E} a_{0}\) is small compared to the fine structure, but take the Lamb \(\operatorname{shift} \delta(\delta=1057 \mathrm{MHz})\) into account (that is, ignore \(2 p_{3 / 2}\) in this calculation). Show that for \(e \mathscr{E} a_{0} \ll \delta\), the energy shifts are quadratic in \(\mathscr{E}\), whereas for \(e \mathscr{E} a_{0} \gg \delta\) they are linear in \(\mathscr{E}\). Briefly discuss the consequences (if any) of time reversal for this problem. This problem is from Gottfried (1966), Problem 7-3.

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.

A \(p\)-orbital electron characterized by \(|n, l=1, m=\pm 1,0\rangle\) (ignore spin) is subjected to a potential $$ V=\lambda\left(x^{2}-y^{2}\right) \quad(\lambda=\text { constant }) $$ a. Obtain the "correct" zeroth-order energy eigenstates that diagonalize the perturbation. You need not evaluate the energy shifts in detail, but show that the original threefold degeneracy is now completely removed. b. Because \(V\) is invariant under time reversal and because there is no longer any degeneracy, we expect each of the energy eigenstates obtained in (a) to go into itself (up to a phase factor or sign) under time reversal. Check this point explicitly.

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?
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