New applications of the formula A |\psi\rangle = \langle A \rangle |\psi\rangle + \Delta A |\psi_{\perp} \rangle are discussed. Simple derivations of the Heisenberg uncertainty principle and of related inequalities are presented. In addition, the formula is used in an instructive paradox which clarifies a fundamental notion in quantum mechanics

Goldenberg, L

Vaidman, L

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Applications of a Simple Formula
Lior Goldenberg and Lev Vaidman
School of Physics and Astronomy
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel-Aviv 69978, Israel.
Abstract
New applications of the formula Aj i = hAij i+ Aj 
?
i are discussed.
Simple derivations of the Heisenberg uncertainty principle and of related in-
equalities are presented. In addition, the formula is used in an instructive
paradox which claries a fundamental notion in quantum mechanics.
1
1 Introduction
The topic of this paper concerns a simple formula, rarely mentioned in the literature,
which can serve as a helpful tool in quantum mechanics. It has been shown [1] that
for any Hermitian operator A and any quantum state j i, the following formula is
valid:
Aj i = hAij i+Aj 
?
i; (1)
where j i, j 
?
i are normalized vectors, h 
?
j i = 0, hAi  h jAj i, and A 
q
hA
2
i   hAi
2
. The proof is as follows: It is always possible to decompose Aj i =
j i+ j 
?
i with   0. Then h jAj i = h j (j i+ j 
?
i) yields  = hAi, and
h jA
y
Aj i = (
?
h j+ 
?
h 
?
j) (j i+ j 
?
i) so that  = A.
In ref. [1] the formula has been applied to a composite system consisting of
a large number of parts in a product state. It was proven that such a product
state is essentially an eigenstate of an operator dened as an \average" of variables
corresponding to these parts. The formula has also been used in a simple derivation
of the minimal time for the evolution of a quantum system to an orthogonal state
[2]. Our aim here is to show new applications of this formula. In section 2 an
immediate result, related to the uncertainty of an operator in two orthogonal states,
is obtained. In section 3 we present an apparent paradox which arises when the
formula is used in a naive way. In section 4 we use it to derive in a simple way the
Heisenberg uncertainty principle and other related inequalities.
2 A Maximal Uncertainty State is Not Unique
Let us rewrite our basic formula in the form
A j i = hAi
 
j i+A
 
j 
?
i: (2)
Then, the scalar product of j 
?
i and A j i is
h 
?
jAj i = A
 
: (3)
2
For A j 
?
i the formula gives
A j 
?
i = hAi
 
?
j 
?
i +A
 
?
j 
??
i: (4)
where h 
??
j 
?
i = 0. Substituting eq.(4) in eq.(3) yields
A
 
= A
 
?
h j 
??
i: (5)
We see that in the case of a two-dimensional Hilbert space where only two mutually
orthogonal states exist, A
 
= A
 
?
. Since jh j 
??
ij  1, eq.(5) leads to
A
 
?
 A
 
: (6)
Thus, we have proved the following theorem:
For any Hermitian operator A and any given state j i there exists a state j 
?
i
orthogonal to j i, such that A
 
?
 A
 
.
This implies that a state corresponding to a maximal uncertainty of any given ob-
servable cannot be unique.
3 An Apparent Paradox
Let us consider a system described by a two-dimensional Hilbert space, such as a
spin-
1
2
particle. Then, for an Hermitian operator B, dierent than A, a relation
similar to eq.(1) holds:
Bj i = hBij i+Bj 
?
i; (7)
where the vector j 
?
i is the same as in eq.(1) (since it is the only vector orthogonal
to j i). Note that the quantities hAi, hBi, A and B are all real numbers.
Multiplying the Hermitian conjugate of eq.(7) by eq.(1), and using the fact that
B = B
y
, we obtain
h jBAj i = (hBih j+Bh 
?
j) (hAij i+Aj 
?
i) = hBihAi +BA: (8)
3
Also, multiplying the Hermitian conjugate of eq.(1) by eq.(7) yields
h jABj i = hAihBi+AB: (9)
Thus, subtracting eq.(8) from eq.(9) leads to
h[A;B]i = 0: (10)
Equation (10) states that the expectation value of the commutator of two arbitrary
operators is always zero, irrespective of the wavefunction which describes the system.
This \remarkable" result can be simply tested using an example of a spin-
1
2
particle, where A = 
x
, B = 
y
, and j i = j "
z
i. The calculation yields
h[A;B]i = h"
z
j[
x
; 
y
]j "
z
i = h"
z
j2 i 
z
j "
z
i = 2 i; (11)
in contradiction to eq.(10).
A paradox? Not really. Obviously, something is wrong in the derivation of
eq.(10). Each of the two basic equations (1) and (7) is separately correct, however,
they are not correct when used together. They both use the same vector j 
?
i
orthogonal to j i, but even in a two-dimensional Hilbert space the orthogonal vector
is uniquely dened only up to a phase. If we use the vector j 
?
i as it is dened by
eq.(1), then we should rewrite eq.(7) as follows:
Bj i = hBij i +B e
i'
j 
?
i; (12)
where ' is real. Equation (10) is now replaced by the correct expression
h[A;B]i = AB (e
i'
  e
 i'
) = 2 iAB sin': (13)
Only if ' = 2n where n = 0;1;2; :::, the expectation value of [A;B] is equal to
zero { not always, as concluded above.
Let us reexamine the previous example with A = 
x
, B = 
y
, and j i = j "
z
i.
Aj i = 
x
j "
z
i = j #
z
i; (14)
Bj i = 
y
j "
z
i = ij #
z
i: (15)
4
Comparing these equations with eqs.(1) and (12) we obtain 
x
= 
y
= 1 and
e
i'
= i (that is sin' = 1). Introducing these values in eq.(13) we nd
h"
z
j[
x
; 
y
]j "
z
i = 2 i; (16)
in perfect agreement with eq.(11). Summing up, the phase in quantum mechanics
is too important to be neglected.
4 The Heisenberg Uncertainty Principle
Our understanding of the preceding apparent paradox has provided us with useful
algebraic tools to be implemented in this section. We present here a simple method,
based on the formula (eq.(1)), to obtain the Heisenberg uncertainty principle. Con-
sider two Hermitian operators, A and B, in an arbitrary Hilbert space. Then, the
following equations hold:
Aj i = hAij i+Aj 
?A
i; (17)
Bj i = hBij i+Bj 
?B
i; (18)
where h 
?A
j i = 0 and h 
?B
j i = 0. Following the same procedure as in section 3,
we nd that
hBAi = hBihAi +BA h 
?B
j 
?A
i; (19)
and
hABi = hAihBi +AB h 
?A
j 
?B
i: (20)
Subtracting eq.(19) from eq.(20) yields
h[A;B]i = AB (h 
?A
j 
?B
i   h 
?B
j 
?A
i)
= 2 iAB Imh 
?A
j 
?B
i: (21)
Then, taking the absolute value of eq.(21) we nd
AB jImh 
?A
j 
?B
ij =
1
2
jh[A;B]ij: (22)
5
The vectors are normalized, therefore jImh 
?A
j 
?B
ij  1, so that we end with
AB 
1
2
jh[A;B]ij; (23)
which is the standard form of the uncertainty principle.
Another interesting inequality can be obtained by calculating the anti-commutator
of A and B. We add eq.(19) to eq.(20) and get
hfA;Bgi = 2 hAihBi +AB (h 
?A
j 
?B
i + h 
?B
j 
?A
i)
= 2 hAihBi + 2AB Reh 
?A
j 
?B
i; (24)
where fA;Bg = AB + BA. Rearranging eq.(24) and taking the absolute values of
both sides, we nd
AB jReh 
?A
j 
?B
ij =




1
2
hfA;Bgi   hAihBi




: (25)
Since jReh 
?A
j 
?B
ij  1, it follows that
AB 




1
2
hfA;Bgi   hAihBi




: (26)
This inequality is a by-product of a conventional derivation of the uncertainty prin-
ciple which is based on the Cauchy-Schwarz inequality [3]. The physical signicance
of eq.(26) is that it provides an estimate for the correlations developed in time be-
tween A and B. For instance, it manifests the correlation between x and p for the
case of a free particle evolving in time [4].
We can also obtain a more accurate expression for AB. Adding eq.(21) to
eq.(24) we nd
h[A;B]i+ hfA;Bgi = 2 iAB Imh 
?A
j 
?B
i
+ 2AB Reh 
?A
j 
?B
i+ 2 hAihBi; (27)
and consequently,
AB h 
?A
j 
?B
i =
1
2
h[A;B]i+
1
2
hfA;Bgi   hAihBi: (28)
6
Taking the norm of both sides we nd
AB 




1
2
h[A;B]i+
1
2
hfA;Bgi   hAihBi




: (29)
Since [A;B] = i C and fA;Bg = D, where C and D are Hermitian operators, and
since the expectation value of an Hermitian operator is a real number, it follows
that
AB 
"

1
2
hfA;Bgi   hAihBi

2
+
1
4
jh[A;B]ij
2
#
1
2
: (30)
This result combines the two previously found bounds, i.e. eqs.(23) and (26).
References
[1] Y. Aharonov and L. Vaidman, \Properties of a Quantum System During the
Time Interval Between Two Measurements", Phys. Rev. A 41, 11 (1990).
[2] L. Vaidman, \Minimal Time for the Evolution to an Orthogonal State", Am.
J. Phys. 60, 182 (1992).
[3] See, for instance, S. Wieder, The Foundations of Quantum Theory (Academic
Press, New York, 1973), pp. 64-65.
[4] D. Bohm, Quantum Theory (Prentice-Hall, Englewood Clis, NJ, 1951), pp.
203-207.
7


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