Adiabatic Phase Shifts for Neutrons and Photons

  • Michael Berry
Part of the NATO ASI Series book series (NSSB, volume 144)


My purpose is to present some new results and make some explanatory remarks about the recently-discovered geometrical phase (Berry 1984) as applied to simple quantal and optical situations. At its most abstract, this phase is a continuation property of the eigenvectors |nx> of a complex Hermitian matrix \( \mathop{H}\limits^{ \wedge } (x) \) depending on (at least two) real parameters X = {X1,X1…}. Let X be taken round a circuit C in parameter space, and let |nx> (assumed nondegenerate) be continued according to the natural transport law
$$ \langle {n_x} | d{n_x}\rangle = 0\ $$
Then |nx> is not a single-valued function of X but acquires round C a phase γn(C) given by the flux through C of a 2-form V, that is
$$ {\gamma_n}(C) = - \iint\limits_S {{V_n}}(X) $$
where S is any surface spanning C. V is given by
$$ {V_n}(X) = {\text{Im }}\langle dn(x) (X)| \wedge |dn(X)\rangle $$
where |n(x)> is any choice of eigenvector which is single-valued over S (of course no such choice satisfies (1)) and ∣dn> is the change in eigenvector resulting from a parameter-space displacement dX. Simon (1983) explained how γn(C) is an example of anholonomy.


Faraday Rotation Geometrical Phase Beam Path Adiabatic Limit Deviative Phase 
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Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • Michael Berry
    • 1
  1. 1.H. H.Wills Physics LaboratoryBristolUK

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