Vector Models in $\mathcal{PT}$ Quantum Mechanics

Abstract

We present two examples of non-Hermitian Hamiltonians which consist of an unperturbed part plus a perturbation that behaves like a vector, in the framework of $\mathcal {PT}$ quantum mechanics. The first example is a generalization of the recent work by Bender and Kalveks, wherein the E2 algebra was examined; here we consider the E3 algebra representing a particle on a sphere, and identify the critical value of coupling constant which marks the transition from real to imaginary eigenvalues. Next we analyze a model with SO(3) symmetry, and in the process extend the application of the Wigner-Eckart theorem to a non-Hermitian setting.

Appendix: Wigner-Eckart Theorem

Suppose we have an angular momentum operator L and a vector operator V satisfying the commutation relations

$$ [ L_i, V_j ] = i \epsilon_{ijk} V_k. $$

(34)

Let |ℓ,m〉 denote an angular momentum multiplet of total angular momentum ℓ and z-component m. Then according to the Wigner-Eckart theorem the matrix elements of V _z and V _±=V _x ±iV _y between multiplet states are determined by the commutation relations Eq. (34). In the usual Wigner-Eckart theorem the Cartesian components of the operator V are assumed to be hermitian. Here we present a non-Hermitian generalization of the theorem.

Following the usual arguments we find the selection rules

(35)

(36)

(37)

Furthermore the matrix elements vanish unless ℓ′=ℓ−1 or ℓ′=ℓ or ℓ′=ℓ+1.

Consider the case ℓ′=ℓ. Generalization of the usual arguments shows that

(38)

where the proportionality constant A is a complex number called the “reduced matrix element”. Note that for V hermitian, A would have to be real, but there is no such restriction in the non-hermitian case.

Similarly in the case ℓ′=ℓ+1 we find

$$ \everymath{\displaystyle} \begin{array} {rcl} \langle\ell+ 1, m + 1 | V_+ | \ell, m \rangle& = & B \biggl[ \frac{ (\ell+ m + 2)( \ell+ m + 1) }{(2 \ell+ 2)(2 \ell+ 1)} \biggr]^{1/2} ,\\[11pt] \langle\ell+ 1, m | V_z | \ell, m \rangle& = & - B \biggl[ \frac{ (\ell- m + 1) ( \ell+ m + 1) }{ (2 \ell+ 2) (2 \ell+ 1) } \biggr]^{1/2} ,\\[11pt] \langle\ell+ 1, m -1 | V_- | \ell, m \rangle& = & - B \biggl[ \frac{ (\ell- m + 1) ( \ell- m + 2 ) }{ (2 \ell+ 2) ( 2 \ell+ 1 ) } \biggr]^{1/2}, \end{array} $$

(39)

where m=−ℓ,…,ℓ and B is another complex reduced matrix element.

Finally in the case that ℓ′=ℓ−1 we find

$$ \everymath{\displaystyle} \begin{array} {rcl} \langle\ell- 1, m + 1 | V_+ | \ell, m \rangle& = & - C \biggl[ \frac{ ( \ell- m - 1)( \ell- m ) }{ (2 \ell) (2 \ell- 1) } \biggr]^{1/2} ,\\[11pt] \langle\ell- 1, m | V_z | \ell, m \rangle& = & - C \biggl[ \frac{ (\ell- m)(\ell+ m) }{ (2 \ell)( 2 \ell- 1) } \biggr]^{1/2} ,\\[11pt] \langle\ell- 1, m - 1 | V_- | \ell, m \rangle& = & C \biggl[ \frac{ (\ell+ m)( \ell+ m - 1 ) }{ (2 \ell)(2 \ell- 1) } \biggr]^{1/2}, \end{array} $$

(40)

where C is a complex reduced matrix element and m=−ℓ,…,ℓ−2 in the first line of Eq. (40), m=−ℓ+1,…,ℓ−1 in the second line of Eq. (40), and m=−ℓ+2,…,ℓ in the last line of Eq. (40).

In the hermitian case the reduced matrix elements satisfy B=C ^∗ but in the non-hermitian case there is no such restriction on the complex elements B and C.