Download Reduction of Uncertainty Relationship For Spin Operator

EJTP 9, No. 26 (2012) 147–152
Electronic Journal of Theoretical Physics
Reduction of Uncertainty Relationship For Spin
Operator
M. A. A. Sbaih∗ , Moeen Kh. Srour† and M. S. Hamada
Department of Physics & Al-Aqsa University Gaza , P.O.Box4501,Gaza,Palestine
Received 20 July 2011, Accepted 7 November 2011, Published 17 January 2012
Abstract: We show in this paper that schrodinger-Robertson Uncertainty relationship for Sx ,
Sy , Sz and for x̂ and p̂ operator can reduced to well known Heisenberg Uncertainty relationship
under suitable transformation.
c Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: ff
PACS (2010):
1.
Introduction
The most Uncertainty relationship, called Schrodinger-Robertson [?, ?] Uncertainty relationship, for two observable  and B̂ satisfying the commutation relationship
Â, B̂] = iĈ
(1)
1
(|F |)2 + (|C|)2
4
(2)
is given by
ΔA2 ΔB 2 ≥
Where ΔA2 and ΔB 2 are the uncertainties associated with  and B̂ respectively, and
defined through
(3)
ΔA2 = A2 − A2
and
F = AB + BA − 2AB
∗
†
Email:mahmoud [email protected]
Email:[email protected]
(4)
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148
is called the first correlation between  and B̂ [?, ?].
The angular bracket in the above equation is the expectation value in the quantum
state. As | F | is non-negative, the schrodinger-Robertson Uncertainty relationship is
often express in the ”Weaker form”
ΔA2 ΔB 2 ≥
1
| C |2
4
(5)
Which is the well known Heisenberg Uncertainty relationship [?] for  and B̂.
The purpose of this paper is to show that it is possible to redefine the observable
without affecting their commutation relation in such way that the first ordered correlation
is zero, and consequently the Schrodinger-Robertson relationship (2) will reduce to the
Heisenberg from (5) itself. We illustrate this for the case of spin as well as for position
and momentum observable.
2.
Spin observable
Given the quantum state, let us first compute the expectation values Sx ,Sy and Sz in some Cartesian coordinate system xyz. These three components gives us the mean
spin vector S, whose polar angle θ and φ with respect to xyz will be
Sx2 + Sy2 −1
θ = tan
(6)
Sz Sy (7)
Sx If now rotated the frame xyz, first rotation about the z − axis through φ and then about
the new y − axis through θ, we obtain a new framex1 y1 z1 whose z1 axis will be along the
mean spin vector S, so that the expectation values
φ = tan−1
Sx1 = Sy1 = 0
Consequently the Schrodinger-Robertson uncertainty relationship in this frame
reduce to
1
| Sx1 Sy1 + Sy1 Sx1 |2 +|Sz |2
ΔSx21 ΔSy21 = ΔSx21 ΔSy21 =
4
(8)
(9)
At this point, we exploit an additional degree of freedom of rotation about the z1 -axis
which along the mean spin vector through an angle α to be chosen later, so that the new
frame is now x2 y2 z2 . As z2 -axis to be alongS, the latter continues to have vanishing
value for x and y component in the new frame too. The spin components in the new
frame are related to old Sx1 , SY1 and SZ1 through
Sx2 = Sx1 cosα + Sy1 sinα
(10)
Electronic Journal of Theoretical Physics 9, No. 26 (2012) 147–152
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Sy2 = −Sx1 sinα + Sy1 cosα
(11)
Sz2 = Sz1
(12)
Sx2 Sy2 + Sy2 Sx2 = Sx1 Sy1 + Sy1 Sx1 cos2α − Sx21 − Sy21 sin2α
(13)
These expression lead to
Here we make the choice
Sx1 Sy1 + Sy1 Sx1 1
α = tan−1
2
Sx21 − Sy21 (14)
so that Sx2 Sy2 + Sy2 Sx2 in equation(13) becomes zero. As a result of which the
Schrodinger-Roberson Uncertainty relationship in equation (2) gets finally reduce to
ΔSx22 ΔSy22 = Sx22 Sy22 ≥
1
| Sz |2
4
(15)
We see that the quantity 14 | Sz |2 is in fact the the greatest lower bound for the product
of the uncertainties in x and y spin components. It is also to be noted here that, the
same result can be formed by rotating the spin state itself.
we will now illustrate this for a spin 1 pure state [?] . A normalized pure state of
a spin 1 system can be expressed in terms of the angular momentum state | 11, | 10,
| 1 − 1 with respect to the axis of a frame xyz as
| ψ = λ1 | 11 + λ0 | 10 + λ−1 | 1 − 1
(16)
with the coefficient satisfying
| λ1 |2 + | λ0 |2 + | λ−1 |2 = 1
(17)
The expectation values Sx , Sy and Sz for this state are given by
1
Sx = √ [λ0 (λ1 + λ−1 ) + λ0 (λ1 + λ−1 )]
2
(18)
i
Sy = √ [λ0 (λ−1 − λ1 ) + λ0 (λ1 − λ−1 )]
2
(19)
Sz =| λ1 |2 − | λ−1 |2
(20)
so that these define the mean spin vector S through
S = Sx î + Sy ĵ + Sz k̂
(21)
The polar angle of this vector with respect to xyz frame can be obtained with the help
of equation(6,7) as
2
| λ0 |2 (| λ1 |2 + | λ−1 |2 ) + λ20 λ−1 λ1 + λ0 λ1 λ−1
√
(22)
tanθ = 2
| λ1 |2 − | λ−1 |2
Electronic Journal of Theoretical Physics 9, No. 26 (2012) 147–152
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tanφ =
i[λ0 (λ−1 − λ1 ) + λ0 (λ1 − λ−1 )]
λ0 (λ−1 + λ1 ) + λ0 (λ1 + λ−1 )
(23)
Applying rotation about the ẑ and ŷ axis, we can transform xyz to x1 y1 z0 which happen
to be a Lakin frame [?], where the expectation value
Sx1 = Sy1 = 0
(24)
Further one can also rotated about ẑ0 axis, conveniently to get the frame x0 y0 z0 such
that state in equation (16) with non zero mean spin value S can be pressed [?] in terms
of the angular momentum basis with respect to ẑ0 as
| Ψ = cos φ | 11z0 + sin φ | 1 − 1z0
(25)
In this frame Sx0 Sy0 + Sy0 Sx0 in equation (25) becomes zero, also the frame Sx0 Sy0 Sz0
which is the Heisenberg uncertainty relationship. It is also to be noted here that only
in the frame x0 y0 z0 obtained from initially chosen xyz through the appropriate rotation
which depend on the spin state itself.
3.
Position and Momentum observable
If we consider the x − p subspace of the phase space of a quantum system [?], we can
make the following three parameter transformation
X = x cos θ + p sin θ + α
(26)
P = −x sin θ + p cos θ + β
(27)
Where α, β are constant, this transformation involves rotation of x and p axes by an
amount θ about an axis perpendicular to the x, p plane and then translation by an
amount α and β respectively along the new position and momentum direction. Now X,P
are the new position and momentum observable related to the old observables x and p.
It is easy to see that the commutation relation is preserved under this transformation i.e
x, p ≡ X, P = i
(28)
Using the transformation, the first ordered correlation of the new observable in any
state can be expressed in term of the old observable as
F new = XP + P X − 2XP F new = F old cos 2θ − Δx2 − Δp2 sin 2θ
So that under the choice
θ=
1
F old
tan−1 2
2
Δx − Δp2
(29)
(30)
(31)
we obtain F new = 0. The Uncertainty relationship X and P then reduce to
1
Δx2 Δp2 ≥ 2
4
(32)
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and 14 2 is then the greatest lower bounded for the product of the new variances.
A glance at the first ordered correlation F old suggests that it becomes zero when
the state is real either in the momentum representation.
4.
Summery
Quantum mechanics is the best fundamental physics theory we have. In this work we had
studies Schrodinger-Robertson uncertainty relationship under suitable transformation for
Sx , Sy and Sz and for x̂ and p̂ operators.
Schrodinger-Robertson under suitable transformation reduce to Heisenberg uncertainty
principle. We can consider Schrodinger-Robertson relationship is a generalization of
Heisenberg uncertainty principle. The the first correlation F always goes to zero under
suitable transformation as shown in the section two and three.