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Two-State System

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

$\displaystyle H_0 \,\vert 1\rangle$ $\displaystyle = E_1 \,\vert 1\rangle,$ (7.2)
$\displaystyle H_0 \,\vert 2\rangle$ $\displaystyle = E_2 \,\vert 2\rangle.$ (7.3)

It is assumed that these states, and their associated eigenvalues, are known. Because $ H_0$ 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

$\displaystyle (H_0 + H_1) \,\vert E\rangle = E\,\vert E\rangle.$ (7.4)

In fact, we can solve this problem exactly. Because the eigenkets of $ H_0$ form a complete set, we can write

$\displaystyle \vert E\rangle = \langle 1\vert E\rangle \vert 1\rangle + \langle 2\vert E\rangle \vert 2\rangle.$ (7.5)

Substituting the previous expansion into Equation (7.4), and then right-multiplying by either $ \langle 1\vert$ or $ \langle 2\vert$ , we get two coupled equations that can be written in matrix form:

$\displaystyle \left( \begin{array}{c c} E_1 -E + e_{11}, & e_{12} \\ e_{12}^{\,...
...ngle\end{array} \!\right)= \left(\!\begin{array}{c}0\\ 0 \end{array}\! \right).$ (7.6)


$\displaystyle e_{11}$ $\displaystyle = \langle 1\vert\,H_1\, \vert 1\rangle,$ (7.7)
$\displaystyle e_{22}$ $\displaystyle = \langle 2 \vert\,H_1\, \vert 2\rangle,$ (7.8)
$\displaystyle e_{12}$ $\displaystyle = \langle 1\vert\,H_1\,\vert 2\rangle$ (7.9)

are the so-called matrix elements of the perturbing Hamiltonian (with respect to the unperturbed eigenstates). In the special (but common) case of a perturbing Hamiltonian whose diagonal matrix elements are zero, so that

$\displaystyle e_{11} = e_{22} = 0,$ (7.10)

the non-trivial solution of Equation (7.6) (obtained by setting the determinant of the matrix equal to zero [92]) is

$\displaystyle E = \frac{(E_1+E_2) \pm \sqrt{(E_1-E_2)^{\,2} + 4\,\vert e_{12}\vert^{\,2}}}{2}.$ (7.11)

Let us expand in the supposedly small parameter

$\displaystyle \epsilon = \frac{\vert e_{12}\vert}{\vert E_1-E_2\vert}.$ (7.12)

We obtain

$\displaystyle E\simeq \frac{1}{2} \,(E_1+E_2) \pm \frac{1}{2}\,(E_1-E_2)\,(1+2\,\epsilon^{\,2} + \cdots).$ (7.13)

The previous expression yields the modifications to the energy eigenvalues caused by the perturbing Hamiltonian:

$\displaystyle E_1'$ $\displaystyle = E_1 + \frac{\vert e_{12}\vert^{\,2}}{E_1-E_2} + \cdots,$ (7.14)
$\displaystyle E_2'$ $\displaystyle = E_2 - \frac{\vert e_{12}\vert^{\,2}}{E_1-E_2} + \cdots.$ (7.15)

Note that $ H_1$ causes the upper eigenvalue to increase, and the lower eigenvalue to decrease. It is easily demonstrated that the modified eigenkets take the form

$\displaystyle \vert 1\rangle'$ $\displaystyle =\vert 1\rangle + \frac{e_{12}^{~\ast}}{E_1-E_2}\, \vert 2\rangle + \cdots,$ (7.16)
$\displaystyle \vert 2\rangle'$ $\displaystyle =\vert 2\rangle - \frac{e_{12}}{E_1-E_2}\, \vert 1\rangle +\cdots.$ (7.17)

Thus, the modified energy eigenstates consist of one of the unperturbed eigenstates with a slight admixture of the other. Actually, the series expansion on the right-hand side of Equation (7.13) only converges when $ 2\,\vert\epsilon\vert<1$ [59]. This suggests that the condition for the validity of the perturbation expansion is

$\displaystyle \vert e_{12}\vert < \frac{\vert E_1-E_2\vert}{2}.$ (7.18)

In other words, when we say that $ H_1$ must be small compared to $ H_0$ , what we really mean is that the previous inequality needs to be satisfied.

next up previous
Next: Non-Degenerate Perturbation Theory Up: Time-Independent Perturbation Theory Previous: Introduction
Richard Fitzpatrick 2016-01-22