91Ó°ÊÓ

The Hamiltonian matrix for a two-state system can be written as $$ \mathscr{H}=\left(\begin{array}{ll} E_{1}^{0} & \lambda \Delta \\ \lambda \Delta & E_{2}^{0} \end{array}\right) $$ Clearly the energy eigenfunctions for the unperturbed problems \((\lambda=0)\) are given by $$ \phi_{1}^{(0)}=\left(\begin{array}{l} 1 \\ 0 \end{array}\right), \quad \phi_{2}^{(0)}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right) . $$ a. Solve this problem exactly to find the energy eigenfunctions \(\psi_{1}\) and \(\psi_{2}\) and the energy eigenvalues \(E_{1}\) and \(E_{2}\). b. Assuming that \(\lambda|\Delta| \ll\left|E_{1}^{0}-E_{2}^{0}\right|\), solve the same problem using time-independent perturbation theory up to first order in the energy eigenfunctions and up to second order in the energy eigenvalues. Compare with the exact results obtained in (a). c. Suppose the two unperturbed energies are "almost degenerate," that is, $$ \left|E_{1}^{0}-E_{2}^{0}\right| \ll \lambda|\Delta| \text {. } $$ Show that the exact results obtained in (a) closely resemble what you would expect by applying degenerate perturbation theory to this problem with \(E_{1}^{0}\) set exactly equal to \(E_{2}^{0}\).

Step by step solution

01

Eigenvalue Problem Setup

To find the energy eigenvalues \(E\), we solve the characteristic equation given by \( \det(\mathscr{H} - E I) = 0 \), where \( I \) is the identity matrix. Substituting \( \mathscr{H} \) into this equation leads to:\[\left| \begin{array}{ll} E_{1}^{0} - E & \lambda \Delta \ \lambda \Delta & E_{2}^{0} - E \end{array} \right| = 0 \] Expanding the determinant gives the equation:\[(E_{1}^{0} - E)(E_{2}^{0} - E) - (\lambda \Delta)^2 = 0 \].

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Key Concepts

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

Energy Eigenvalues

In the context of quantum mechanics, energy eigenvalues play a pivotal role. An energy eigenvalue is simply the energy level that a quantum system can be found in. When we solve the Schrödinger equation for a Hamiltonian operator, the solutions yield both the possible energy states (eigenvalues) and the corresponding states or functions (eigenfunctions).

For the exercise problem given, the Hamiltonian matrix for a two-state system is used to find these energy values. The Hamiltonian includes parameters such as the unperturbed energies \(E_1^0\) and \(E_2^0\), and the term \(\lambda \Delta\) which introduces a perturbation.

The determinant method shows how the energy eigenvalues, \(E_1\) and \(E_2\), are derived. You expand the determinant of \(\mathscr{H} - EI\) and set it to zero. This results in a quadratic equation in \(E\), reflecting that there are two unique energy states for the two-state system. Even when perturbations are introduced, the goal is to solve for \(E\) such that the determinant remains zero. This ensures that the derived energies are valid states the system can take.

Perturbation Theory

Perturbation theory is a valuable tool in quantum mechanics, particularly when dealing with systems that can be seen as slight modifications of a simpler problem. It helps solve systems where an exact analytic solution is either very difficult or impossible.

In our exercise, perturbation theory is employed by assuming \(\lambda |\Delta| \ll |E_1^0 - E_2^0|\). This means the perturbation is small compared to the separation between unperturbed energy levels. First-order perturbation theory provides corrections to the eigenfunctions, while second-order perturbation theory offers corrections to the eigenvalues.

By applying these methods, we adjust our energy levels and eigenfunctions to better approximate the real system after the perturbation is applied. The technique assumes the system's initial or "unperturbed" solutions as a starting point. It then incrementally adjusts these solutions to account for the perturbation. In practice, this results in more accurate models for certain quantum systems without requiring a full solution of the new problem.

Two-State System

A two-state system, often a vital component in quantum mechanics studies, is the simplest model system where a quantum object, such as an atom or a fermion, can exist in two possible states or levels. This system allows us to clearly understand the fundamental principles of quantum behavior.

In this exercise, the Hamiltonian matrix for a two-state system provides a structure to find energy eigenvalues and eigenfunctions. Here, our two possible states are represented by the basis functions \(\phi_1^{(0)}\) and \(\phi_2^{(0)}\). These states correspond to the system's initial "unperturbed" condition when there is no interaction between states — represented mathematically as \(\lambda = 0\).

The beauty of the two-state system model is its simplicity, yet it perfectly illustrates crucial quantum concepts. Even with a basic Hamiltonian matrix, we can observe quantum phenomena such as energy level splitting due to interactions between states. This simple and elegant model can be extended and applied to understand more complex systems in quantum optics, solid-state physics, and other fields.