- Email: [email protected]
A note on the uncertainty relation between the position and momentum
s __
25 November 1996
I@ A
PHYSICS
__
LETTERS
A
i
Physics Letters A 223 (1996)9- 11
ELSEWER
A note on the uncertainty relation between the position and momentum Ting Yu ’ Theory Group, Blackeri Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ UK Received
14 February
1996; revised manuscript received 23 September 1996; accepted Communicated by P.R. Holland
for publication
24 September
1996
Abstract In this note a derivation of the parameter-based uncertainty relation between position and momentum proposed by Uffink and Hilgevoord is given. This uncertainty relation can be regarded as an exact counterpart of the time-energy uncertainty relation.
In Ref. [ l] Mandelstam and Tamm prove the following uncertainty relation between the time and energy
for
O
?rfi 2Afi’
where AH = dm. Since then the parameter based uncertainty relation has been extensively studied in various contexts [2-111. For an excellent review, e.g., see Ref. [ 81. The parameter based uncertainty relations between position and momentum have been given in Refs. [5,7]. A particular interesting case can be formulated as follows: For a given state represented by I@(X)), we use 6x to denote the distance for the translation of I@(X)) to an orthogonal state and AP to denote the momentum uncertainty in the state I@(x)), then ’ E-mail: [email protected]. 037%9601/96/$12.00 Copyright PII SO375-960 1(96)00744-X
where h is the Planck constant without being divided by 27~. The interpretation of (2) may be as follows: The distance for a displaced observer who “sees” the quantum state is completely different from that the undisplaced observer sees, being not less than AP/fi. Note that this uncertainty relation is on the same footing as that between time and energy (see Refs. [ 4,5,7] >. In this note we shall provide a different derivation for inequality (2), we also prove that the right hand side of the above inequality cannot be improved, namely, h/4 is the greatest lower bound. Our derivation is analogous to that in Ref. [lo] for the timeenergy uncertainty relation. In the following, for simplicity, we only consider the wave functions with one spatial dimension in position space. Also, we assume that the system is not in an eigenstate of the momentum operator P. In order to get the shortest distance for the translation to an orthogonal state we introduce a formal quantity that will play a central role in our derivation.
0 1996 Elsevier Science B.V. All rights reserved.
IO
T. YU / Physics Letters A 223 (1996) 9-1 I
This quantity is known as the correlation amplitude. To begin with, let us consider a system in a state described by the wave function II/J(X)) (with a slight abuse of notation), we use the notation I$( x -u)) for the displaced state. The correlation amplitude associated with these two states is defined as C(a)
= ($(x,19(x
- a,).
(3)
The modulus of C(u) provides a quantitative measure of the “resemblance” between the states which are localised around different places, or equivalently, if the state remains unchanged but the observer is displaced, then the above “between the states” should be replaced by “between the observers”. In what follows we will find the smallest D which satisfies C(D)
= (ifi(x)l$(x
- D)) = 0.
(4)
It is convenient to use ]C(u) 1’. Taking the derivative with respect to the parameter a, we get the following equation, $lC(o)/z
= 2Re ((+(x
Using (6)) we obtain
$lti(x -u)) = -$P),$(x_u,) +
A@&
- u))l,
where p is the momentum operator. Combining with the identity (7), it follows that
$lC(u)12 =-““,” --Re[i(lCl(x
this
- u)l@(x))
x (Icl(x)l@cII(x - a))].
(10)
In order to get the desired minimum distance, we wish to make the function IC(u) J? decrease to zero as fast as possible. For given Ap and 1C(a) 1, we have to find themaximum value of I(J/(x)l$~(x - a))1 Next consider the expansion of the initial state
MX))Y I@(x)) = wcx - ~)l$(x))l@(x - a,) + ((cli(x - a)lIcl(X))l~l(x - u,) + Al$II(.~ - a,).
(11)
where ($(x - u)~@J_(.Y - u)) = 0 and ($_L(x a) jtil_~ (X-U)) = 0. This leads to the followingequation,
- a)l@(.r))
I(SCx-4w~P + lAl2.
1= Let 1$(x-u)) fl,
(9)
be the state displaced over the distance
+ I(J/I(X - W,(*))l’ (12)
That is I$( X - u,) = e-(ilQPGli@ X)).
(6) 1($1(x
It follows easily that
$1$(x - a)) = For any Hermitian holds [ 11,121, h Al+(x))
*
=
- IC(u)l2
- Ihl.
(13)
formula (14)
1
@Ml(x)) + AAkb~(x))>
(8)
Now we consider the following equation,
$(u)i’= --2~~c(u),~rjmj?(15)
where (A^) = (fi(x)l&(x)), and i+(x))
= Jl
Obviously, I($1 (x - a) lrl/(x)) I attains its maximum when h = 0. Thus, the maximal possible value of the rate of change of IC( a) I2 is given by
+ic”cx- u)). operator d, the following
- a)llCl(x))i
A.R = dm
and II,!J’_L(x))satisfy
(lcl(.r, I$1 (x1) = 0.
Let us introduce a new parameter lC(u)l, then Eq. (14) reads L
8, and let sin6’ =
(16)
T. Yu/Phyks
Letters A 223 (19961 9-11
4 denotes the derivative with respect to the parameter CI. 6 = rr and n-/2 correspond to the orthogonal state and the initial state, respectively. It follows that
Dd-
(17)
4AP”
where D is the shortest distance to an orthogonal state. Suppose we have a physical system whose momentum uncertainty is As. Let us denote the distance for the translation to an orthogonal state by 6x, then Ax is not less than 11/4Aa, namely IGAB 3 f.
(18)
This completes the proof of ( I). So far we have worked exclusively in one-space for simplicity, but everything we have done can be generalised to threespace. In summary, let us emphasise that the usual position-momentum uncertainty relation is on a quite different footing from that for time and energy. In some sense. the uncertainty relation (17) can be regarded as an exact counterpart of the time-energy uncertainty relation. To prove that /1/4 on the right hand side of (1) cannot be improved, we consider a simple example. Suppose a physical system whose state is represented by the wave function
where N is the normalisation constant. We use 6x and A.? to denote the distance of the translation to an orthogonal state and the uncertainty of the position operator 4, respectively. It is easy to see that 6x goes to 7r/2 as n ---f cx), but AP goes to infinity. This example shows that the uncertainty relation (18) is of quite different nature from the usual uncertainty rclation between the position and momentum. It is easy to see that if AP goes to fi, then &A~ goes to h/4. This indicates that D = h/4 is not only a lower bound but also the greatest lower bound or the infimum. The author is grateful to Professor Z. Wu for interesting discussions.
References [I
I
L. Mandelstam
if -2nrr
= Ne-(-r-ZnV)‘sin
121 G.N.
Fleming, Nuovo
13 I K. Bhattacharyya, 141 J. Uffink
= Ne-(x+2n7d
if 2mr
< x < +CQ,
sin x.
if -oc
< x < -2i17r, (19)
J. Phys.
(USSR)
9 ( 1945)
J. Hilgevoord
( 1985)
263. 925.
59.
( 1985)
Phys. L.ett. A I I I
and J. tiffink.
formalism.
Dordrecht, J.
A I6 (1973)
Found. Phys. IS (1985)
Phys. Lett. A I08
161 J.M. Levy-Leblond, quantum
Cimento
J. Phys. A 16 ( 1983) 2993.
and J. Hilgevoord.
[S] J. Uffmk,
in: Microphysical
eds. A. van der Menve
3.53. reality
and
et al. (Kluwcr.
1988).
Hilgevoord
Uncertainty,
and
J.
ed. A. Miller
Uffink,
in:
(Plenum.
Sixty-two New York,
years 1990).
of and
references therein.
< x < 2n7r,
x,
and 1. Tamm,
249.
J. Anandan
(Cltl(.r) = Nsinx,
II
and Y. Aharonov.
Phys. Rev. Lett.
65
( 1991)
1697. [lo]
L. Vaidman,
[ II I I 12 ]
J. Uffmk,
Am. J. Phys. 60
( 1992)
Am. J. Phys. 61 (1993)
Y. Aharonov
and L. Vaidman.
182.
935.
Phys. Rev. A 41 ( 1990)
I I.
25 November 1996
I@ A
PHYSICS
__
LETTERS
A
i
Physics Letters A 223 (1996)9- 11
ELSEWER
A note on the uncertainty relation between the position and momentum Ting Yu ’ Theory Group, Blackeri Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ UK Received
14 February
1996; revised manuscript received 23 September 1996; accepted Communicated by P.R. Holland
for publication
24 September
1996
Abstract In this note a derivation of the parameter-based uncertainty relation between position and momentum proposed by Uffink and Hilgevoord is given. This uncertainty relation can be regarded as an exact counterpart of the time-energy uncertainty relation.
In Ref. [ l] Mandelstam and Tamm prove the following uncertainty relation between the time and energy
for
O
?rfi 2Afi’
where AH = dm. Since then the parameter based uncertainty relation has been extensively studied in various contexts [2-111. For an excellent review, e.g., see Ref. [ 81. The parameter based uncertainty relations between position and momentum have been given in Refs. [5,7]. A particular interesting case can be formulated as follows: For a given state represented by I@(X)), we use 6x to denote the distance for the translation of I@(X)) to an orthogonal state and AP to denote the momentum uncertainty in the state I@(x)), then ’ E-mail: [email protected]. 037%9601/96/$12.00 Copyright PII SO375-960 1(96)00744-X
where h is the Planck constant without being divided by 27~. The interpretation of (2) may be as follows: The distance for a displaced observer who “sees” the quantum state is completely different from that the undisplaced observer sees, being not less than AP/fi. Note that this uncertainty relation is on the same footing as that between time and energy (see Refs. [ 4,5,7] >. In this note we shall provide a different derivation for inequality (2), we also prove that the right hand side of the above inequality cannot be improved, namely, h/4 is the greatest lower bound. Our derivation is analogous to that in Ref. [lo] for the timeenergy uncertainty relation. In the following, for simplicity, we only consider the wave functions with one spatial dimension in position space. Also, we assume that the system is not in an eigenstate of the momentum operator P. In order to get the shortest distance for the translation to an orthogonal state we introduce a formal quantity that will play a central role in our derivation.
0 1996 Elsevier Science B.V. All rights reserved.
IO
T. YU / Physics Letters A 223 (1996) 9-1 I
This quantity is known as the correlation amplitude. To begin with, let us consider a system in a state described by the wave function II/J(X)) (with a slight abuse of notation), we use the notation I$( x -u)) for the displaced state. The correlation amplitude associated with these two states is defined as C(a)
= ($(x,19(x
- a,).
(3)
The modulus of C(u) provides a quantitative measure of the “resemblance” between the states which are localised around different places, or equivalently, if the state remains unchanged but the observer is displaced, then the above “between the states” should be replaced by “between the observers”. In what follows we will find the smallest D which satisfies C(D)
= (ifi(x)l$(x
- D)) = 0.
(4)
It is convenient to use ]C(u) 1’. Taking the derivative with respect to the parameter a, we get the following equation, $lC(o)/z
= 2Re ((+(x
Using (6)) we obtain
$lti(x -u)) = -$P),$(x_u,) +
A@&
- u))l,
where p is the momentum operator. Combining with the identity (7), it follows that
$lC(u)12 =-““,” --Re[i(lCl(x
this
- u)l@(x))
x (Icl(x)l@cII(x - a))].
(10)
In order to get the desired minimum distance, we wish to make the function IC(u) J? decrease to zero as fast as possible. For given Ap and 1C(a) 1, we have to find themaximum value of I(J/(x)l$~(x - a))1 Next consider the expansion of the initial state
MX))Y I@(x)) = wcx - ~)l$(x))l@(x - a,) + ((cli(x - a)lIcl(X))l~l(x - u,) + Al$II(.~ - a,).
(11)
where ($(x - u)~@J_(.Y - u)) = 0 and ($_L(x a) jtil_~ (X-U)) = 0. This leads to the followingequation,
- a)l@(.r))
I(SCx-4w~P + lAl2.
1= Let 1$(x-u)) fl,
(9)
be the state displaced over the distance
+ I(J/I(X - W,(*))l’ (12)
That is I$( X - u,) = e-(ilQPGli@ X)).
(6) 1($1(x
It follows easily that
$1$(x - a)) = For any Hermitian holds [ 11,121, h Al+(x))
*
=
- IC(u)l2
- Ihl.
(13)
formula (14)
1
@Ml(x)) + AAkb~(x))>
(8)
Now we consider the following equation,
$(u)i’= --2~~c(u),~rjmj?(15)
where (A^) = (fi(x)l&(x)), and i+(x))
= Jl
Obviously, I($1 (x - a) lrl/(x)) I attains its maximum when h = 0. Thus, the maximal possible value of the rate of change of IC( a) I2 is given by
+ic”cx- u)). operator d, the following
- a)llCl(x))i
A.R = dm
and II,!J’_L(x))satisfy
(lcl(.r, I$1 (x1) = 0.
Let us introduce a new parameter lC(u)l, then Eq. (14) reads L
8, and let sin6’ =
(16)
T. Yu/Phyks
Letters A 223 (19961 9-11
4 denotes the derivative with respect to the parameter CI. 6 = rr and n-/2 correspond to the orthogonal state and the initial state, respectively. It follows that
Dd-
(17)
4AP”
where D is the shortest distance to an orthogonal state. Suppose we have a physical system whose momentum uncertainty is As. Let us denote the distance for the translation to an orthogonal state by 6x, then Ax is not less than 11/4Aa, namely IGAB 3 f.
(18)
This completes the proof of ( I). So far we have worked exclusively in one-space for simplicity, but everything we have done can be generalised to threespace. In summary, let us emphasise that the usual position-momentum uncertainty relation is on a quite different footing from that for time and energy. In some sense. the uncertainty relation (17) can be regarded as an exact counterpart of the time-energy uncertainty relation. To prove that /1/4 on the right hand side of (1) cannot be improved, we consider a simple example. Suppose a physical system whose state is represented by the wave function
where N is the normalisation constant. We use 6x and A.? to denote the distance of the translation to an orthogonal state and the uncertainty of the position operator 4, respectively. It is easy to see that 6x goes to 7r/2 as n ---f cx), but AP goes to infinity. This example shows that the uncertainty relation (18) is of quite different nature from the usual uncertainty rclation between the position and momentum. It is easy to see that if AP goes to fi, then &A~ goes to h/4. This indicates that D = h/4 is not only a lower bound but also the greatest lower bound or the infimum. The author is grateful to Professor Z. Wu for interesting discussions.
References [I
I
L. Mandelstam
if -2nrr
= Ne-(-r-ZnV)‘sin
121 G.N.
Fleming, Nuovo
13 I K. Bhattacharyya, 141 J. Uffink
= Ne-(x+2n7d
if 2mr
< x < +CQ,
sin x.
if -oc
< x < -2i17r, (19)
J. Phys.
(USSR)
9 ( 1945)
J. Hilgevoord
( 1985)
263. 925.
59.
( 1985)
Phys. L.ett. A I I I
and J. tiffink.
formalism.
Dordrecht, J.
A I6 (1973)
Found. Phys. IS (1985)
Phys. Lett. A I08
161 J.M. Levy-Leblond, quantum
Cimento
J. Phys. A 16 ( 1983) 2993.
and J. Hilgevoord.
[S] J. Uffmk,
in: Microphysical
eds. A. van der Menve
3.53. reality
and
et al. (Kluwcr.
1988).
Hilgevoord
Uncertainty,
and
J.
ed. A. Miller
Uffink,
in:
(Plenum.
Sixty-two New York,
years 1990).
of and
references therein.
< x < 2n7r,
x,
and 1. Tamm,
249.
J. Anandan
(Cltl(.r) = Nsinx,
II
and Y. Aharonov.
Phys. Rev. Lett.
65
( 1991)
1697. [lo]
L. Vaidman,
[ II I I 12 ]
J. Uffmk,
Am. J. Phys. 60
( 1992)
Am. J. Phys. 61 (1993)
Y. Aharonov
and L. Vaidman.
182.
935.
Phys. Rev. A 41 ( 1990)
I I.