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Chapter 7: Problem 13

In the text I asserted that the first-order corrections to an \(n\) -fold degenerate energy are the eigenvalues of the \(W\) matrix, and I justified this claim as the "natural" generalization of the case \(n=2 .\) Prove it, by reproducing the steps in Section \(7.2 .1,\) starting with $$\psi^{0}=\sum_{j=1}^{n} \alpha_{j} \psi_{j}^{0}$$ (generalizing Equation 7.17 ), and ending by showing that the analog to Equation 7.27 can be interpreted as the eigenvalue equation for the matrix \(\mathrm{W}\)

Short Answer

Expert verified
The first-order corrections are the eigenvalues of the matrix \(W\).

Step by step solution

01

Expand the Perturbative Wave Function

Begin by expanding the zero-order wave function, denoted as \( \psi^0 \), in terms of a linear combination of the degenerate zero-order eigenstates \( \psi_j^0 \):\[\psi^0 = \sum_{j=1}^{n} \alpha_j \psi_j^0.\]Each \( \alpha_j \) represents a coefficient in this linear combination.
02

Apply the Time-Independent Perturbation Theory

Use the time-independent Schr枚dinger equation for a perturbed system:\[(H^0 + \lambda W) \psi = E \psi,\]where \(H^0\) is the Hamiltonian of the unperturbed system, \(W\) is the perturbation, \(\psi\) is the perturbed wave function, and \(E\) is the total energy including corrections.
03

Substitute and Simplify

Substitute the expansion from Step 1 into the perturbed Schr枚dinger equation:\[(H^0 + \lambda W) \sum_{j=1}^{n} \alpha_j \psi_j^0 = E \sum_{j=1}^{n} \alpha_j \psi_j^0.\]Focus specifically on the terms involving \( H^0 \) and use the fact that each \( \psi_j^0 \) is an eigenfunction of \( H^0 \), giving:\[H^0 \psi_j^0 = E^0 \psi_j^0.\]Thus, the expression simplifies to:\[\lambda \sum_{j=1}^{n} \alpha_j W \psi_j^0 = \sum_{j=1}^{n} \alpha_j E^1 \psi_j^0.\]
04

Formulate the Matrix Equation

Multiply through by the adjoint of \(\psi_i^0\) and integrate over all space, making use of orthonormality:\[\sum_{j=1}^{n} \alpha_j \langle \psi_i^0 | W | \psi_j^0 \rangle = E^1 \alpha_i.\]This can be rewritten as a matrix equation \( \mathbf{W} \mathbf{\alpha} = E^1 \mathbf{\alpha} \), where \( \mathbf{W} \) is a matrix with elements \( W_{ij} = \langle \psi_i^0 | W | \psi_j^0 \rangle \).
05

Eigenvalue Problem Interpretation

Recognize this as an eigenvalue problem for the matrix \( \mathbf{W} \):\[\mathbf{W} \mathbf{\alpha} = E^1 \mathbf{\alpha}.\]The first-order energy correction \( E^1 \) is thus the eigenvalue of the matrix \( \mathbf{W} \) corresponding to the vector \( \mathbf{\alpha} \).

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Key Concepts

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

Degenerate Perturbation Theory
In quantum mechanics, degenerate perturbation theory is a refined technique used when dealing with systems that have degenerate energy levels. Degeneracy means that there are multiple wave functions corresponding to the same energy level. When a small perturbation, or disturbance, is introduced into such a system, the degenerate levels can split or shift. Ordinary perturbation theory is not adequate here because it would not account for the complexity of these changes.
The process starts by expanding the wave function of the system into a linear combination of its degenerate eigenfunctions. Each wave function is associated with a coefficient that adjusts based on the influence of the perturbation. This method allows finding how these coefficients change due to the perturbation, leading to a new matrix equation that helps to determine the system's new energy levels.
  • This theory generalizes non-degenerate perturbation theory by accounting for any number of degenerate states.
  • It's essential for understanding complex quantum systems, like atoms and molecules with multiple electrons.
By solving the matrix equation, physicists can calculate the first-order energy corrections, allowing a better understanding of the impact of small disturbances on quantum systems.
Eigenvalue Equation
The eigenvalue equation is a key mathematical expression in quantum mechanics, especially used to determine energy levels in a quantum system. It involves finding the scalar value鈥攌nown as the eigenvalue鈥攖hat characterizes how an eigenvector is stretched or compressed by a matrix.
In the context of degenerate perturbation theory, the eigenvalue equation is expressed as \( \mathbf{W} \mathbf{\alpha} = E^1 \mathbf{\alpha} \). Here:
  • \(\mathbf{W}\) is the matrix containing perturbation effects.
  • \(\mathbf{\alpha}\) is the eigenvector of coefficients for the linear combination of wave functions.
  • \(E^1\) is the eigenvalue representing the first-order correction to the energy.
This formulation shows that the energy corrections due to perturbation are closely tied to the eigenvalues of the perturbation matrix. Solving this eigenvalue problem is crucial because:
  • It reveals the new energy levels of the system.
  • Helps in predicting how the system would behave under slight perturbations.
Understanding and solving the eigenvalue equation is fundamental in analyzing quantum systems.
Schr枚dinger Equation
The Schr枚dinger equation is the cornerstone of quantum mechanics, describing how quantum systems evolve over time. For stable systems, the time-independent form is extensively used and pertinent when dealing with perturbation theories. This form looks like \((H^0 + \lambda W) \psi = E \psi\), where:
  • \(H^0\) is the original, unperturbed Hamiltonian operator.
  • \(\lambda W\) represents the perturbation potential being applied.
  • \(\psi\) is the wave function of the system.
  • \(E\) is the total energy, including perturbation effects.
The role of the Schr枚dinger equation in perturbation theory is to determine how the wave function鈥攁nd hence the quantum state鈥攔esponds to small disturbances. In finding solutions to this equation for a perturbed system:
  • Systems with Hamiltonians that include small perturbations can have significant impacts on physical predictions, especially for degenerate systems.
  • The perturbative approach allows these effects to be systematically calculated, providing insights into complex atomic and molecular structures.
Grasping the Schr枚dinger equation and its applications in fluctuating scenarios is pivotal for anyone looking to delve into deeper nuances of quantum physics.
Time-Independent Perturbation Theory
Time-independent perturbation theory is a mathematical approach in quantum mechanics used to investigate systems where a small, constant disturbance is introduced. Unlike time-dependent theories, this approach examines how the system's energy levels and states are modified without considering time as a variable. This is an essential tool for:
  • Explaining the fine structure of atomic spectra due to electron interactions.
  • Determining energy shifts in response to external fields, which are constant in time.
The methodology involves expanding the wave function into a series, taking the perturbation into account as an additional term added to the system's Hamiltonian. Key steps include:
  • Using the Schr枚dinger equation to include the perturbation in calculations.
  • Solving the resulting equations to find altered energies and new wave functions.
The strength of time-independent perturbation theory lies in its ability to provide accurate predictions about quantum systems with degenerate states. It is crucial for analyzing the impact of slight but steady influences on atomic and molecular systems, expanding the horizons of quantum mechanics beyond what unperturbed models can achieve.

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Most popular questions from this chapter

Apply perturbation theory to the most general two-level system. The unperturbed Hamiltonian is \(\mathrm{H}^{0}=\left(\begin{array}{cc}E_{a}^{0} & 0 \\ 0 & E_{b}^{0}\end{array}\right)\) and the perturbation is \(\mathrm{H}^{\prime}=\lambda\left(\begin{array}{ll}V_{a a} & V_{a b} \\ V_{b a} & V_{b b}\end{array}\right)\) with \(V_{b a}=V_{a b}^{*}, V_{a a}\) and \(V_{b b}\) real, so that \(\mathrm{H}\) is hermitian. As in Section \(7.1 .1, \lambda\) is a constant that will later be set to 1. (a) Find the exact energies for this two-level system. (b) Expand your result from (a) to second order in \(\lambda\) (and then set \(\lambda\) to 1 ). Verify that the terms in the series agree with the results from perturbation theory in Sections 7.1 .2 and \(7.1 .3 .\) Assume that \(E_{b}>E_{a}\) (c) Setting \(V_{a a}=V_{b b}=0,\) show that the series in (b) only converges if $$\left|\frac{V_{a b}}{E_{b}^{0}-E_{a}^{0}}\right| < \frac{1}{2}$$ Comment: In general, perturbation theory is only valid if the matrix clements of the perturbation are small compared to the energy level spacings. Otherwise, the first few terms (which are all we ever calculate) will give a poor approximation to the quantity of interest and, as shown here, the series may fail to converge at all, in which case the first few terms tell us nothing.

Evaluate the following commutators: (a) \([\mathbf{L} \cdot \mathbf{S}, \mathbf{L}],(\mathbf{b})[\mathbf{L} \cdot \mathbf{S}, \mathbf{S}]\) \((\mathrm{c})[\mathbf{L} \cdot \mathbf{S}, \mathbf{J}],(\mathrm{d})\left[\mathbf{L} \cdot \mathbf{S}, L^{2}\right],(\mathrm{e})\left[\mathbf{L} \cdot \mathbf{S}, S^{2}\right],(\mathrm{f})\left[\mathbf{L} \cdot \mathbf{S}, J^{2}\right] .\) Hint: \(\mathbf{L}\) and \(\mathbf{S}\) satisfy. the fundamental commutation relations for angular momentum (Equations 4.99 and 4.134 ), but they commute with each other.

Estimate the correction to the ground state energy of hydrogen due to the finite size of the nucleus. Treat the proton as a uniformly charged spherical shell of radius \(b,\) so the potential energy of an electron inside the shell is constant: \(-e^{2} /\left(4 \pi \epsilon_{0} b\right) ;\) this isn't very realistic, but it is the simplest model, and it will give us the right order of magnitude. Expand your result in powers of the small parameter ( \(b / a\) ), where \(a\) is the Bohr radius, and keep only the leading term, so your final answer takes the form $$\frac{\Delta E}{E}=A(b / a)^{n}$$ Your business is to determine the constant \(A\) and the power \(n\). Finally, put in \(b \approx 10^{-15} \mathrm{m}\) (roughly the radius of the proton) and work out the actual number. How does it compare with fine structure and hyperfine structure?

Consider a quantum system with just three linearly independent states. Suppose the Hamiltonian, in matrix form, is $$\mathrm{H}=V_{0}\left(\begin{array}{ccc} (1-\epsilon) & 0 & 0 \\ 0 & 1 & \epsilon \\ 0 & \epsilon & 2 \end{array}\right)$$ where \(V_{0}\) is a constant, and \(\epsilon\) is some small number \((\epsilon \ll 1)\) (a) Write down the eigenvectors and eigenvalues of the unperturbed Hamiltonian \((\epsilon=0)\) (b) Solve for the exact eigenvalues of \(H\) Expand each of them as a power series in \(\epsilon,\) up to second order. (c) Use first- and second-order non-degenerate perturbation theory to find the approximate eigenvalue for the state that grows out of the nondegenerate eigenvector of \(\mathrm{H}^{0}\). Compare the exact result, from (b). (d) Use degenerate perturbation theory to find the first-order correction to the two initially degenerate eigenvalues. Compare the exact results.

By appropriate modification of the hydrogen formula, determine the hyperfine splitting in the ground state of (a) muonic hydrogen (in which a muon-same charge and \(g\) -factor as the electron, but 207 times the mass substitutes for the electron), (b) positronium (in which a positron-same mass and \(g\) -factor as the electron, but opposite charge- substitutes for the proton), and (c) muonium (in which an anti-muon-same mass and \(g\) -factor as a muon, but opposite charge-substitutes for the proton). Hint: Don't forget to use the reduced mass (Problem 5.1) in calculating the "Bohr radius" of these exotic "atoms," but use the actual masses in the gyromagnetic ratios. Incidentally, the answer you get for positronium \(\left(4.82 \times 10^{-4} \mathrm{eV}\right)\) is quite far from the experimental value \(\left(8.41 \times 10^{-4} \mathrm{eV}\right) ;\) the large discrepancy is due to pair annihilation \(\left(e^{+}+e^{-} \rightarrow \gamma+\gamma\right),\) which contributes an extra (3/4) \(\Delta E,\) and does not occur (of course) in ordinary hydrogen, muonic hydrogen, or muonium.
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