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Physics LibreTexts

7.2: Two-State System

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Let us start by considering time-independent perturbation theory, in which the modification to the Hamiltonian, H1 , has no explicit dependence on time. It is usually assumed that the unperturbed Hamiltonian, H0 , is also time-independent.

Consider the simplest non-trivial system, in which there are only two independent eigenkets of the unperturbed Hamiltonian. These are denoted

H0|1 = E1|1, ??? H0|2 = E2|2. ???


It is assumed that these states, and their associated eigenvalues, are known. Because H0 is, by definition, an Hermitian operator, its two eigenkets are mutually orthogonal and form a complete set. The lengths of these eigenkets are both normalized to unity. Let us now try to solve the modified energy eigenvalue problem

(H0+H1)|E=E|E. ???

In fact, we can solve this problem exactly. Since the eigenkets of H0 form a complete set, we can write

|E=1|E|1+2|E|2. ???

Right-multiplication of Equation ??? by 1| and 2| yields two coupled equations, which can be written in matrix form:

$ \left( \begin{array}{c c} E_1 -E + e_{11} & e_{12} \\ e_{12}^{\,\...
...ngle\end{array} \!\right)= \left(\!\begin{array}{c}0\\ 0 \end{array}\! \right).$ ???

Here,

e11 = 1|H1|1, ??? e22 = 2|H1|2, ??? e12 = 1|H1|2. ???


In the special (but common) case of a perturbing Hamiltonian whose diagonal matrix elements (in the unperturbed eigenstates) are zero, so that

e11=e22=0, ???

the solution of Equation ??? (obtained by setting the determinant of the matrix equal to zero) is

E=(E1+E2)±(E1E2)2+4|e12|22. ???

Let us expand in the supposedly small parameter

ϵ=|e12||E1E2|. ???

We obtain

E12(E1+E2)±12(E1E2)(1+2ϵ2+). ???

The above expression yields the modifications to the energy eigenvalues due to the perturbing Hamiltonian:

E1 = E1+|e12|2E1E2+, ??? E2 = E2|e12|2E1E2+. ???


Note that H1 causes the upper eigenvalue to rise, and the lower eigenvalue to fall. It is easily demonstrated that the modified eigenkets take the form

|1 = |1+e 12E1E2|2+, ??? |2 = |2e12E1E2|1+. ???


Thus, the modified energy eigenstates consist of one of the unperturbed eigenstates with a slight admixture of the other. Note that the series expansion in Equation ??? only converges if 2|ϵ|<1 . This suggests that the condition for the validity of the perturbation expansion is

|e12|<|E1E2|2. ???

In other words, when we say that H1 needs to be small compared to H0 , what we really mean is that the above inequality needs to be satisfied.

Contributors

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 7.2: Two-State System is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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