It is shown that the exact coherent states for the time-dependent harmonic oscillator constructed by Hartley and Ray are equivalent to the well-known squeezed states.Some time ago Hartley and ~a~, ' in a very interesting I I n , t ) = f i ( n + f ) I n , t ) .( 1 1 ) paper, constructed exact coherent states for the timedependent harmonic oscillator by making use of the Lewis-Riesenfeld quantum theory for the time-dependent harmonic o~c i l l a t o r .~ According to Hartley and Ray, these "new" coherent states have all the properties of the coherent states for the time-independent oscillator3 except that the uncertainty product of position and momentum is not minimum; i.e., they are not minimum-uncertainty states. The purpose of the present paper is to show that the "new" coherent states constructed by these authors are equivalent to the well-known squeezed statesAp6In what follows I present a brief summary of Ref. 1 . Consider the time-dependent harmonic oscillator where q,p are canonically conjugate and w ( t ) is the timedependent harmonic-oscillator frequency. An invariant for the Hamiltonian ( 1 ) is given by'92 Then, using the Lewis-Riesenfeld theory,2 Hartley and Ray constructed coherent states for the time-dependent oscillator ( 1 ) . These states are given by ' where a is a complex number and the phase functions a , ( t are given by 1 dt' a , ( t ) = -( n + + ) J -0 p 2 ( t ' ) The subscript S in ( 1 2 ) indicates that the states evolve in time according to the Schrodinger equation. The coherent states / a , t )s are eigenstates of the operator b ( t ) with eigenvalue cr(t): where where q ( t ) satisfies the harmonic-oscillator equation I dt' a o ( t ) = -+J -q + w 2 ( t ) q = o (3) 0 p 2 ( t ' ) and p ( t ) is any solution of the auxiliary equation Now, by calculating the uncertainty in q and p in the state / a , t ) s , one finds 3 p + w Z ( t ) p = 1 / p .(4) fi ( A q ) 2 = -p 2 , Now, by using the time-dependent operatorslx2 2 fi (~p ) ' = -( p ' + l / p 2 ) .2 So, the uncertainty relation is expressed as and, in general, does not attain its minimum value. we can rewrite the invariant ( 2 ) as As remarked by Hartley and Ray, all of their resultsfor the Hamiltonian ( 1 ) reduce to the usual time-( 7 ) independent oscillator in the limit w(t)-wO=const if we The operators b ( t and b ' ( t ) have the properties take the particular solution b t ( t ) I n , t ) = ( n + 1 ) 1 / 2 1 n + l , t ) , for the auxiliary equation (4). In fact, in this case the ( I 0 ) operator b i t ) transforms into the ordinary annihilation where the states n,t ) are eigenstates of the invariant I, operator for the time-independent oscillator [see Eq. ( 2 1 ) i.e., below]. Also, observe that relation ( 1 9 ) reduces to 36 1279 -
