Must a Hamiltonian be Hermitian?

American Journal of Physics

Volumn 71, Issue 11, 2003, Pages 1095-1102

  Bender, Carl M ^a   Brody, Dorje C ^b   Jones, Hugh F ^b  

^a WASHINGTON UNIVERSITY   (United States)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0242307914     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1574043     Document Type: Article

References (46)

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7. -t do not exhibit this time-reversal symmetry while the solution y(t) = cosh(t) is time-reversal symmetric. The same is true with a system whose Hamiltonian is PT symmetric. Even if the Schrödinger equation and the corresponding boundary conditions are PT symmetric, the wave function that solves the Schrödinger equation boundary value problem may not be symmetric under space-time reflection. When the solution exhibits PT symmetry, we say that the PT symmetry is unbroken. Conversely, if the solution does not possess PT symmetry, we say that the PT symmetry is broken.
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27. The PT norm of a state determines its parity type (Ref. 4). We can regard C as representing the operator that determines the C charge of the state. Quantum states having opposite C charge possess opposite parity type.
28. The parity operator in coordinate space is explicitly real P(x,y) = δ(x + y); the operator C(x,y) is complex because it is a sum of products of complex functions, as we see in (15). The complexity of the C operator can be seen explicitly in perturbative calculations of C(x,y) (Ref. 21).
29. Note that if a function satisfies a linear ordinary differential equation, then the function is analytic wherever the coefficient functions of the differential equation are analytic. The Schrödinger equation (4) is linear and its coefficients are analytic except for a branch cut at the origin; this branch cut can be taken to run up the imaginary axis. We choose the integration contour for the inner product (8) so that it does not cross the positive imaginary axis. Path independence occurs because the integrand of the inner product (8) is a product of analytic functions.
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46. Rotating from θ=0 to θ=-π, we obtain the same Hamiltonian as in (32) but the spectrum is the complex conjugate of the spectrum obtained when we rotate from θ=0 to θ=π.