- Email: [email protected]
Local values in quantum mechanics
8 April 1996 PHYSICS
LETTERS
A
Physics Letters A 212(1996) 315-319
ELSEVIER
Local
in Leon Cohen Iv2
Hunter
College and Graduate
Center of CUNE
695 Park Avenue, New York, NY 10021.
USA
Received 24 January 1995; revised manuscript received 9 January 1996; accepted for publication 19 January 1996 Communicated by P.R. Holland
Abstract An exact equation is derived for the uncertainty of an operator in terms of the real and imaginary parts of &c/l*. We show that the uncertainty always consists of two terms which are qualitatively different and that can be interpreted in terms of local values. Exact equations are given for the spread of momentum, energy, and squeezing operator in terms of the amplitude and phase of the wave function.
2
,.
1. Introduction (A&2=
The standard quantum mechanical current and the Bohm quantum potential are examples of local quantities. Our aim is to obtain a general expression for
the local value of an arbitrary physical variable. For the sake of clarity we first present our results where “local” means spatial locality and subsequently generalize to variables other than spatial. Our main result is as follows. For a wave function, (I,(4)) and an observable represented by the operator A, we first break up the quantity A+/$ into its real and imaginary parts,
y
l+(q)12dq ) I
0
+/-
[(~)R-(6)12,~(q),2d (1.2)
We will then argue that the natural interpretation of this equation is that the local value of the observable, and its local spread, denoted by (A), and a& respectively, are
(A),=
($ 1 )
( 1.3)
R
(1.1)
and (1.4)
We will prove the following exact expression for the
uncertainty of A,
’ E-mail: [email protected]. *Work supported in part by the NSA HBCU/MI program and the PSC-CUNY Research Award Program.
The possibility of interpreting Eq. (1.3) as a local value has been previously proposed [ I-31 from different viewpoints and the recent outstanding book by Holland [2] discusses this issue and gives a number of examples.
037%9601/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved PII SO375-9601(96)00075-8
316
L. Cohen/Physics
Letters A 212 (1996) 315-319
2. The uncertainty of an operator
2 ralb
We now derive Eq. (1.2) and show how one can interpret it in terms of local values, The standard deviation is
=
=
s
@*(9)(2 - (&)2t,U9) d9
s
I(A - (&$(9)12
d9
(2.3)
g&P(b)
((a)b - (a))‘P(b)
db +
J
(2.1) (2.2)
b) da.
(2.9)
Now, one can show that the standard deviation and conditional standard deviation are related by [ 41 a,2 =
(A&2 =
& /(a - (u)12P(a,
db,
s
(2.10) where P(b) is the marginal distribution of b. Comparison of Eq. (2.10) with Eq. (2.5) suggests the expressions ( 1.3) and ( 1.4). Of course it is improper to uniquely equate functions under the integral sign, however the identification as given leads to plausible results. 3. Examples
Since we are considering Hermitian operators, (d) is real. Hence, the first two terms of the integrand in Eq. (2.4) are real and therefore
3.1. Momentum and quantum mechanical current
For the operator A we take the momentum operator p = (li/i)d/dq. Accordingly we have (3.1)
(3.2)
(2.5)
This equation is exact. It expresses the standard deviation as the sum of two quantities. Note that both terms are averages, the second term being the average of a quantity subtracted from the global mean. To understand the nature of this equation and the significance of the two terms consider a probability distribution of two variables a and b which we denote by P (a, b) . The average value of a and the conditional value of a for a fixed b are respectively
=;
[($$4) +(g$+)‘]
+i$
[(-j-f:+)
(F>,
= &;($.(9)y
($%)r aP( a, b) da. (a)/? = P(b) s
(a))2P(a,
>
7
(3.4)
b) dadb,
= -$5($*(9)d5$
(2.7) +ti(9)dq
Also, the standard deviation of a and the conditional standard deviation are
JJ(a-
d+* (9)
(2.6)
1
fl.2 ‘I =
(3.3)
Therefore
-Jl(s)dq aP(a, 6) dadb,
- (fF!9+)*].
(2.8)
de* (9) >
.
(3.5)
Alternatively, and more revealing, we first express the wave function in terms of its amplitude and phase e/(q) = Z?(q)eisCq)‘“,
(3.6)
L. Cohen/Physics
317
Letters A 212 (19%) 315-319
Therefore, if the time dependent Schrodinger equation is imposed, then
then (3.7)
-i/i,’ --”
”
ICI at
$
and hence
=S’(q)
(3.8)
7
1 JR =-i/5x,,-,,
as
(3.
(;)‘/(S’+2F)iR’(q)dq
(A&2=
(3. 6)
=-
hR’(q)
(3.9)
R(q)’
&St2 +
+
V(q)
Substituting these values in F!q, (1.2) we have -
$$
-
= fi2
f;
(H))2R2(q)
dq
(3.17)
2
+
I
[S’(q) -
b)12R2(q)
dq.
S(
(3.10)
This is an exact equation for the uncertainty of momentum expressed in terms of the phase and amplitude of the wave function. From Eqs. (1.3) and ( 1.4) we have [l-5]
>
R2(q)dq
(3.18)
+/($+(H))2R2~qNq.
Using Eq. ( 1.3) we obtain [ 2,5] (A),=
(!2),= v(q)+$s~2-;;.(3.19)
The last term is the Bohm quantum potential [ 6,7]. The conditional standard deviation is (3.11)
a$,=
($>,=(E>’(6’+*g2.
(3.12) Eq. (3.11) is recognized to be the quantum mechanical current per unit mass per unit density.
(3.20) Also, (3.21)
3.2. Energy, local energy and the quantum potential 2
(3.22)
We consider the Hamiltonian A=;+“(q),
(3.13)
and operating on the wave function as given by JZq. (3.6) we have
3.3. Example: squeezing operator
The compression operator is
c= p&+P&),
(3.23)
which gives (3.14)
(3.24)
318
L. Cohen/
Physics Letters A 212 (1996) 315-319
4.1. Average value of position for a given value of momentum
Therefore R’(q) qR(q)
(AC)2 = /i2
+
s
(@‘(q)-
1 2 + 5 R2(s) dq >
(C))2 R2(s> dq.
(3.25)
We take A = $ = ilid/dp, that is the position operator in the momentum representation and for B we take the momentum, p. Denoting the momentum wave function by 4(p) we have that
and the local value of c is 0,
(3.26)
= @‘(q),
which is position times the quantum rent.
mechanical
=-&;
cur-
(4J*(P)y (4.5)
Alternatively if we express the momentum tion in terms of its phase and amplitude
4. Local value of an observable for a fixed value of another observable and further examples Suppose that in calculating the spread of an operator we use the wave function in the b representation, v (6)) instead of the position representation. The spread is therefore
4(p)
wave func-
= T(p)eiU@)l”,
(4.6)
then
04 -_LE&p~-K!& - U’(p). -= 4
(A&2
q*(b)(d
=
- (A))2~(b)
db,
(4.1)
s where the operator A^must be expressed in the variables of the b representation. Structurally this equation is no different than Eq. (2.2) and hence the identical steps leading to Eq. (1.2) now lead to
(4.7)
Using Eq. ( 1. I ) we have (ri,,
(4.8)
= -U’(P)
and the local standard deviation &2
2 u41P
T’(P) ( To’
is
2
(4.9)
>
4.2. Example: energy for a given momentum In the momentum
A=
c+
V(-Gig).
representation (4.10)
which suggests taking Therefore
(A)p
() ()
$ ,
(4.3)
R
2
(+&
=
!!I! q
(4.4)
1’
IQ, (4.4) gives the “local” value of the observable for a fixed value, b. We now consider some examples.
A
4.3. Position for a given harmonic oscillator energy The wave function in the Harmonic oscillator energy representation is the set of coefficients, {cn), where
319
L. Cohen/ Physics Letters A 212 (1996) 31‘5-319
VW43 t) =
&mn(q)
(4.12)
and where u, are the harmonic oscillator eigenfunctions. Now, the position operator, & in the harmonic energy representation, is
Since the operator is Hermitian the eigenvalue is real, in which case, according to Eqs. ( 1.3) and ( 1.4))
(A),=
($ ) A* 2=o (1
(5.2)
=a,
R
(1 + A,)-‘, (4.13) where cy = mw/ri and A is the differencing operator, AC, = C,+I - c,. Using the fact that ( 1 + A,)m c, = c,,+,, we have that
a& = *
.
(5.3)
1
Second, we expect the average value to be the local value averaged over all space,
(A) = /(&#I*
dq = 1
($)
[r//l* dq.
(5.4)
R Therefore
This is seen to be the case since .
we have that
(A^/=/+*&dq=/$$12dq (4.15) To consider a concrete example take the wave function ‘P exp[-+*(q-
so)*],
(4.16)
o”4”o
exp[-i(n
=I [($)R+i ($)I lr1/1*dq
(5.6)
=
(5.7)
where the second term in Eq. (5.6) is zero because we know that the average value of a Hermitian operator is real. For an arbitrary representation we have
for which c, = exp(-+*qi)-
(5.5)
+ ~)&i~t].
&G2 (4.17) Using these values on obtains
(A)= /(&lrl(b)l*db = (5.8)
(4.18)
and the proof is identical to that just given.
5. Conclusion
References
In conclusion we make two comments. First, we point out that if the wave function happens to be an eigenfunction of the operator then Eq. ( 1.3) predicts that the local value is a constant throughout space and is given by the eigenvalue and furthermore the local spread as given by Eq. ( 1.4) is zero. This can be seen as follows. For an eigenstate, u,(q) where ,&(q) = au,,(q), we have
[I] L. Cohen and C. Lee, Found. Phys. 17 (1987) 561.
&I(q)/&l(q)
= a.
(5.1)
[2] P.R. Holland, The quantum theory of motion (Cambridge Univ. Press, Cambridge, 1993). ]3] K.K. Wan and P Sumner, Phys. Len. A 128 (1988) 458. [4] L. Cohen, Found. Phys. 20 (1990) 1455. [5] L. Cohen, in: Foundations of quantum mechanics, eds. T.D. Black, M.M. Nieto, H.S. Pilloff, M.O. Scully and R.M. Sinclair (World Scientific, Singapore, 1992) p. 78. [6] D. Bohm, Phys. Rev. 85 (1952) 166. [7] D. Bohm, Phys. Rev. 85 ( 1952) 180.
LETTERS
A
Physics Letters A 212(1996) 315-319
ELSEVIER
Local
in Leon Cohen Iv2
Hunter
College and Graduate
Center of CUNE
695 Park Avenue, New York, NY 10021.
USA
Received 24 January 1995; revised manuscript received 9 January 1996; accepted for publication 19 January 1996 Communicated by P.R. Holland
Abstract An exact equation is derived for the uncertainty of an operator in terms of the real and imaginary parts of &c/l*. We show that the uncertainty always consists of two terms which are qualitatively different and that can be interpreted in terms of local values. Exact equations are given for the spread of momentum, energy, and squeezing operator in terms of the amplitude and phase of the wave function.
2
,.
1. Introduction (A&2=
The standard quantum mechanical current and the Bohm quantum potential are examples of local quantities. Our aim is to obtain a general expression for
the local value of an arbitrary physical variable. For the sake of clarity we first present our results where “local” means spatial locality and subsequently generalize to variables other than spatial. Our main result is as follows. For a wave function, (I,(4)) and an observable represented by the operator A, we first break up the quantity A+/$ into its real and imaginary parts,
y
l+(q)12dq ) I
0
+/-
[(~)R-(6)12,~(q),2d (1.2)
We will then argue that the natural interpretation of this equation is that the local value of the observable, and its local spread, denoted by (A), and a& respectively, are
(A),=
($ 1 )
( 1.3)
R
(1.1)
and (1.4)
We will prove the following exact expression for the
uncertainty of A,
’ E-mail: [email protected]. *Work supported in part by the NSA HBCU/MI program and the PSC-CUNY Research Award Program.
The possibility of interpreting Eq. (1.3) as a local value has been previously proposed [ I-31 from different viewpoints and the recent outstanding book by Holland [2] discusses this issue and gives a number of examples.
037%9601/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved PII SO375-9601(96)00075-8
316
L. Cohen/Physics
Letters A 212 (1996) 315-319
2. The uncertainty of an operator
2 ralb
We now derive Eq. (1.2) and show how one can interpret it in terms of local values, The standard deviation is
=
=
s
@*(9)(2 - (&)2t,U9) d9
s
I(A - (&$(9)12
d9
(2.3)
g&P(b)
((a)b - (a))‘P(b)
db +
J
(2.1) (2.2)
b) da.
(2.9)
Now, one can show that the standard deviation and conditional standard deviation are related by [ 41 a,2 =
(A&2 =
& /(a - (u)12P(a,
db,
s
(2.10) where P(b) is the marginal distribution of b. Comparison of Eq. (2.10) with Eq. (2.5) suggests the expressions ( 1.3) and ( 1.4). Of course it is improper to uniquely equate functions under the integral sign, however the identification as given leads to plausible results. 3. Examples
Since we are considering Hermitian operators, (d) is real. Hence, the first two terms of the integrand in Eq. (2.4) are real and therefore
3.1. Momentum and quantum mechanical current
For the operator A we take the momentum operator p = (li/i)d/dq. Accordingly we have (3.1)
(3.2)
(2.5)
This equation is exact. It expresses the standard deviation as the sum of two quantities. Note that both terms are averages, the second term being the average of a quantity subtracted from the global mean. To understand the nature of this equation and the significance of the two terms consider a probability distribution of two variables a and b which we denote by P (a, b) . The average value of a and the conditional value of a for a fixed b are respectively
=;
[($$4) +(g$+)‘]
+i$
[(-j-f:+)
(F>,
= &;($.(9)y
($%)r aP( a, b) da. (a)/? = P(b) s
(a))2P(a,
>
7
(3.4)
b) dadb,
= -$5($*(9)d5$
(2.7) +ti(9)dq
Also, the standard deviation of a and the conditional standard deviation are
JJ(a-
d+* (9)
(2.6)
1
fl.2 ‘I =
(3.3)
Therefore
-Jl(s)dq aP(a, 6) dadb,
- (fF!9+)*].
(2.8)
de* (9) >
.
(3.5)
Alternatively, and more revealing, we first express the wave function in terms of its amplitude and phase e/(q) = Z?(q)eisCq)‘“,
(3.6)
L. Cohen/Physics
317
Letters A 212 (19%) 315-319
Therefore, if the time dependent Schrodinger equation is imposed, then
then (3.7)
-i/i,’ --”
”
ICI at
$
and hence
=S’(q)
(3.8)
7
1 JR =-i/5x,,-,,
as
(3.
(;)‘/(S’+2F)iR’(q)dq
(A&2=
(3. 6)
=-
hR’(q)
(3.9)
R(q)’
&St2 +
+
V(q)
Substituting these values in F!q, (1.2) we have -
$$
-
= fi2
f;
(H))2R2(q)
dq
(3.17)
2
+
I
[S’(q) -
b)12R2(q)
dq.
S(
(3.10)
This is an exact equation for the uncertainty of momentum expressed in terms of the phase and amplitude of the wave function. From Eqs. (1.3) and ( 1.4) we have [l-5]
>
R2(q)dq
(3.18)
+/($+(H))2R2~qNq.
Using Eq. ( 1.3) we obtain [ 2,5] (A),=
(!2),= v(q)+$s~2-;;.(3.19)
The last term is the Bohm quantum potential [ 6,7]. The conditional standard deviation is (3.11)
a$,=
($>,=(E>’(6’+*g2.
(3.12) Eq. (3.11) is recognized to be the quantum mechanical current per unit mass per unit density.
(3.20) Also, (3.21)
3.2. Energy, local energy and the quantum potential 2
(3.22)
We consider the Hamiltonian A=;+“(q),
(3.13)
and operating on the wave function as given by JZq. (3.6) we have
3.3. Example: squeezing operator
The compression operator is
c= p&+P&),
(3.23)
which gives (3.14)
(3.24)
318
L. Cohen/
Physics Letters A 212 (1996) 315-319
4.1. Average value of position for a given value of momentum
Therefore R’(q) qR(q)
(AC)2 = /i2
+
s
(@‘(q)-
1 2 + 5 R2(s) dq >
(C))2 R2(s> dq.
(3.25)
We take A = $ = ilid/dp, that is the position operator in the momentum representation and for B we take the momentum, p. Denoting the momentum wave function by 4(p) we have that
and the local value of c is 0,
(3.26)
= @‘(q),
which is position times the quantum rent.
mechanical
=-&;
cur-
(4J*(P)y (4.5)
Alternatively if we express the momentum tion in terms of its phase and amplitude
4. Local value of an observable for a fixed value of another observable and further examples Suppose that in calculating the spread of an operator we use the wave function in the b representation, v (6)) instead of the position representation. The spread is therefore
4(p)
wave func-
= T(p)eiU@)l”,
(4.6)
then
04 -_LE&p~-K!& - U’(p). -= 4
(A&2
q*(b)(d
=
- (A))2~(b)
db,
(4.1)
s where the operator A^must be expressed in the variables of the b representation. Structurally this equation is no different than Eq. (2.2) and hence the identical steps leading to Eq. (1.2) now lead to
(4.7)
Using Eq. ( 1. I ) we have (ri,,
(4.8)
= -U’(P)
and the local standard deviation &2
2 u41P
T’(P) ( To’
is
2
(4.9)
>
4.2. Example: energy for a given momentum In the momentum
A=
c+
V(-Gig).
representation (4.10)
which suggests taking Therefore
(A)p
() ()
$ ,
(4.3)
R
2
(+&
=
!!I! q
(4.4)
1’
IQ, (4.4) gives the “local” value of the observable for a fixed value, b. We now consider some examples.
A
4.3. Position for a given harmonic oscillator energy The wave function in the Harmonic oscillator energy representation is the set of coefficients, {cn), where
319
L. Cohen/ Physics Letters A 212 (1996) 31‘5-319
VW43 t) =
&mn(q)
(4.12)
and where u, are the harmonic oscillator eigenfunctions. Now, the position operator, & in the harmonic energy representation, is
Since the operator is Hermitian the eigenvalue is real, in which case, according to Eqs. ( 1.3) and ( 1.4))
(A),=
($ ) A* 2=o (1
(5.2)
=a,
R
(1 + A,)-‘, (4.13) where cy = mw/ri and A is the differencing operator, AC, = C,+I - c,. Using the fact that ( 1 + A,)m c, = c,,+,, we have that
a& = *
.
(5.3)
1
Second, we expect the average value to be the local value averaged over all space,
(A) = /(&#I*
dq = 1
($)
[r//l* dq.
(5.4)
R Therefore
This is seen to be the case since .
we have that
(A^/=/+*&dq=/$$12dq (4.15) To consider a concrete example take the wave function ‘P exp[-+*(q-
so)*],
(4.16)
o”4”o
exp[-i(n
=I [($)R+i ($)I lr1/1*dq
(5.6)
=
(5.7)
where the second term in Eq. (5.6) is zero because we know that the average value of a Hermitian operator is real. For an arbitrary representation we have
for which c, = exp(-+*qi)-
(5.5)
+ ~)&i~t].
&G2 (4.17) Using these values on obtains
(A)= /(&lrl(b)l*db = (5.8)
(4.18)
and the proof is identical to that just given.
5. Conclusion
References
In conclusion we make two comments. First, we point out that if the wave function happens to be an eigenfunction of the operator then Eq. ( 1.3) predicts that the local value is a constant throughout space and is given by the eigenvalue and furthermore the local spread as given by Eq. ( 1.4) is zero. This can be seen as follows. For an eigenstate, u,(q) where ,&(q) = au,,(q), we have
[I] L. Cohen and C. Lee, Found. Phys. 17 (1987) 561.
&I(q)/&l(q)
= a.
(5.1)
[2] P.R. Holland, The quantum theory of motion (Cambridge Univ. Press, Cambridge, 1993). ]3] K.K. Wan and P Sumner, Phys. Len. A 128 (1988) 458. [4] L. Cohen, Found. Phys. 20 (1990) 1455. [5] L. Cohen, in: Foundations of quantum mechanics, eds. T.D. Black, M.M. Nieto, H.S. Pilloff, M.O. Scully and R.M. Sinclair (World Scientific, Singapore, 1992) p. 78. [6] D. Bohm, Phys. Rev. 85 (1952) 166. [7] D. Bohm, Phys. Rev. 85 ( 1952) 180.