aa r X i v : . [ qu a n t - ph ] A ug On Minimum Uncertainty States
Pankaj Sharan
Physics Department, Jamia Millia Islamia, New Delhi, 110 025, India
Abstract
Necessary and sufficient condition for the existence of a minimumuncertainty state for an arbitrary pair of observables is given.
Let the states of a physical system be represented by normalized vectorsin a Hilbert space H . For two vectors φ and ψ in H , denote the innerproduct by ( ψ, φ ) and define the norm k φ k of φ by k φ k = ( φ, φ ). Let A and B be two observables; that is, self-adjoint operators. Let the observable C be defined by the commutator [ A, B ] = iC . The expectation value ( ψ, Aψ )of A is denoted by a . Similarly, expectation values of B and C in the state ψ are denoted b and c respectively.The statement of the uncertainty inequality is∆ A ∆ B ≥ | c | , (1)where the variance (or uncertainty) of A in the state ψ is defined as ∆ A = k ( A − a ) ψ k and a similar formula for ∆ B . We say that ψ is a minimumuncertainty state (MUS) for the pair A, B if the equality is achieved in (1)above, that is, if ∆ A ∆ B = 12 | c | . (2)The proof of the uncertainty inequality is a direct application of theSchwarz inequality which states that | ( ψ, φ ) | ≤ k ψ kk φ k (3)for any two vectors φ and ψ in H . We assume that one of the vectors (say φ )is non-zero to avoid triviality. The Schwarz inequality becomes an equalityif and only if ψ can be written as the other (non-zero) vector φ multipliedby a complex number z ψ = zφ. (4)1he proof of the uncertainty inequality is as follows. Denote by Im z theimaginary part of a complex number z . The Schwarz inequality implies∆ A ∆ B = k ( A − a ) ψ kk ( B − b ) ψ k≥ | (( A − a ) ψ, ( B − b ) ψ ) | Ineqauality 1 ≥ |