262
J. Moran and V. Hussin
Using these transformations we can compute the dispersions in x
(Δ ˆ
X)
2
|Ξ α,β
=
1
2
+ |α|
2 sinh
2 R + Re(e
iΘ α
2 ) sinh R cosh R;
(Δ ˆ
P x )
2
|Ξ α,β
=
1
2
+ |α|
2 sinh
2 R − Re(e
iΘ α
2 ) sinh R cosh R,
(25)
and similarly for y
(Δ ˆ
Y )
2
|Ξ α,β
=
1
2
+ |β|
2 sinh
2 R + Re(e
iΘ β
2 ) sinh R cosh R;
(Δ ˆ
P y )
2
|Ξ α,β
=
1
2
+ |β|
2 sinh
2 R − Re(e
iΘ β
2 ) sinh R cosh R.
(26)
These results also hold for the generalized squeezed states |Ψ, Ξ α,β because the
action of the displacement operator has no effect on the variances. The results
resemble those in Eq. (5) but are modified by α, β. We see in the limit R → 0
we saturate the Heisenberg uncertainty relation in both x and y.
6 Conclusion
In this paper we have described a method for constructing squeezed states for the
2D isotropic oscillator which relies on using the minimal set of definitions used to
describe the squeezed states of the 1D oscillator. Unlike the coherent states defined
in a similar manner in [1], the generalized squeezed states did not factorize into the
product of squeezed states on x and y independently. A coupling was induced which
took the form of a two-mode like squeezing creating an entanglement between the
two modes.
We found the dispersions for the 2D squeezed states and saw that they resemble
the dispersions in the 1D case but modified by the parameters α, β introduced during
the construction of the SU (2) coherent states. As well we saw a separation of
the spatial probability densities into two distinct maxima for larger values of the
squeezing R.
Finally, perhaps this method can be used to construct squeezed states for more
general degenerate and higher dimensional systems and oscillators. The approach
presented in this paper will require modification on a case by case basis because in
general a multidimensional system will admit a more complex degenerate structure,
which would significantly modify the generalized ladder operators as well as the
non-degenerate basis {|ν}. If a system possesses non-algebraic degeneracies, such
as the 2D particle in a box (e.g., 1 2 + 7 2 = 5 2 + 5 2 ), a new method for counting
states contributing to a degenerate subgroup |ν would be required.
J. Moran and V. Hussin
Using these transformations we can compute the dispersions in x
(Δ ˆ
X)
2
|Ξ α,β
=
1
2
+ |α|
2 sinh
2 R + Re(e
iΘ α
2 ) sinh R cosh R;
(Δ ˆ
P x )
2
|Ξ α,β
=
1
2
+ |α|
2 sinh
2 R − Re(e
iΘ α
2 ) sinh R cosh R,
(25)
and similarly for y
(Δ ˆ
Y )
2
|Ξ α,β
=
1
2
+ |β|
2 sinh
2 R + Re(e
iΘ β
2 ) sinh R cosh R;
(Δ ˆ
P y )
2
|Ξ α,β
=
1
2
+ |β|
2 sinh
2 R − Re(e
iΘ β
2 ) sinh R cosh R.
(26)
These results also hold for the generalized squeezed states |Ψ, Ξ α,β because the
action of the displacement operator has no effect on the variances. The results
resemble those in Eq. (5) but are modified by α, β. We see in the limit R → 0
we saturate the Heisenberg uncertainty relation in both x and y.
6 Conclusion
In this paper we have described a method for constructing squeezed states for the
2D isotropic oscillator which relies on using the minimal set of definitions used to
describe the squeezed states of the 1D oscillator. Unlike the coherent states defined
in a similar manner in [1], the generalized squeezed states did not factorize into the
product of squeezed states on x and y independently. A coupling was induced which
took the form of a two-mode like squeezing creating an entanglement between the
two modes.
We found the dispersions for the 2D squeezed states and saw that they resemble
the dispersions in the 1D case but modified by the parameters α, β introduced during
the construction of the SU (2) coherent states. As well we saw a separation of
the spatial probability densities into two distinct maxima for larger values of the
squeezing R.
Finally, perhaps this method can be used to construct squeezed states for more
general degenerate and higher dimensional systems and oscillators. The approach
presented in this paper will require modification on a case by case basis because in
general a multidimensional system will admit a more complex degenerate structure,
which would significantly modify the generalized ladder operators as well as the
non-degenerate basis {|ν}. If a system possesses non-algebraic degeneracies, such
as the 2D particle in a box (e.g., 1 2 + 7 2 = 5 2 + 5 2 ), a new method for counting
states contributing to a degenerate subgroup |ν would be required.