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Weigert, S. orcid.org/0000-0002-6647-3252 (1996) How to determine a quantum state by
measurements: The Pauli problem for a particle with arbitrary potential. Physical Review A.
pp. 2078-2083. ISSN 1094-1622
https://doi.org/10.1103/PhysRevA.53.2078
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How to determine a quantum state by measurements: The Pauli problem for a particle
with arbitrary potential
Stefan Weigert
Institut fu
¨
r Physik der Universita
¨
t Basel Klingelbergstra
b
e 82, CH-4056 Basel, Switzerland
~Received 13 June 1995!
The problem of reconstructing a pure quantum state
u
c
&
from measurable quantities is considered for a
particle moving in a one-dimensional potential V(x). Suppose that the position probability distribution
u
c
(x,t)
u
2
has been measured at time t, and let it have M nodes. It is shown that after measuring the time
evolved distribution at a short-time interval Dt later,
u
c
(x,t1 Dt)
u
2
, the set of wave functions compatible with
these distributions is given by a smooth manifold M in Hilbert space. The manifold M is isomorphic to an
M -dimensional torus, T
M
. Finally, M additional expectation values of appropriately chosen nonlocal operators
fix the quantum state uniquely. The method used here is the analog of an approach that has been applied
successfully to the corresponding problem for a spin system.
PACS number~s!: 03.65.Sq
I. INTRODUCTION
In this paper progress is reported concerning a deceptively
simple question known as the Pauli problem: does the mea-
surement of the probability densities for position and mo-
mentum of a particle determine its quantum state? Originat-
ing from a footnote in Pauli’s article in Handbuch der Physik
@1#, this question has led, in a more general setting, to a
number of investigations over the past decades: the expecta-
tion values of which sets of operators characterize uniquely a
~pure or mixed! state of a quantum system? Apparently, one
important early motivation for dealing with this problem has
been to demystify the concept of the wave function @2#: be-
ing a complex quantity it seems impossible to directly ob-
serve it in experiments. However, if an appropriate set of
expectation values provides the same information about a
quantum system as does the wave function itself, then it is
reasonable to consider the wave function just as a particu-
larly convenient description of the system.
Various works investigating the Pauli problem have been
reviewed in @3~a!#. There are many possibilities to approach
the problem in its general form since one is free to choose
the set of operators to be measured at will. It has been shown
by various authors @2,4– 6# that knowledge of position and
momentum distributions alone ~being equivalent to knowl-
edge of the expectation values of all powers of position and
momentum operators, respectively! does not single out one
specific state. This result is obtained from explicitly con-
structing states with identical probability distributions of
both position and momentum. It is not known, in general,
how large such a set of ‘‘Pauli partners’’ actually is, and
which supplementary expectation values would allow one to
distinguish between them. For more details and a list of ref-
erences the reader is referred to @3~a!#.
A novel approach is based on tomographic methods in
order to determine the Wigner function of a quantum state
@7,8#. In its sequel, successful experimental realizations have
been reported. The quantum state of an electromagnetic field
mode has been reconstructed by using a method called ‘‘op-
tical homodyne tomography’’ @9#. It relies on the possibility
of measuring probability distributions of the quadrature am-
plitude x
ˆ
w
5 x
ˆ
cos
w
1p
ˆ
sin
w
, obtained from rotating the posi-
tion operator in the phase plane by an angle
w
. In a similar
vein, the vibrational quantum state of a molecule has been
determined through the measurement of a time-dependent
spectrum of fluorescence @10#. The information obtained in
this way from ‘‘molecular emission tomography’’ can be
shown to encode a quasiprobability function, and it allows
one thus to reconstruct the sought-after quantum state. A re-
lated proposal has been made to experimentally determine
the state of both a scalar light wave or a particle wave-
function @11#. It exploits the transformations induced on the
Wigner function when the state under investigation passes
through well-defined lenses while propagating along some
direction in space.
It is straightforward to pose this problem for a quantum
spin of length s. The setting in a finite-dimensional Hilbert
space turns out to be an important modification. Using a
specific version of a Stern-Gerlach apparatus ~in which indi-
vidual beams after the separation can be shielded and the
remaining ones can be brought together again! allows one to
measure directly the intensities and relative phases of the
splitted beams @12#. It is not clear, however, whether it is
actually possible to perform this experiment without destroy-
ing the phase relations. It has been shown in @3~a!# that mea-
suring the intensities of the components of a spin state along
two ~infinitesimally! close directions in space is compatible
with 2
2s
spin states. In this setup the familiar version of the
Stern-Gerlach apparatus is sufficient. It turns out that all
Pauli partners can be exhibited explicitly, and it is sufficient
to know the expectation value of one additional well-defined
operator in order to uncover the actual quantum-mechanical
state. This procedure is not constructive: uniqueness of the
state compatible with the measurements is shown—its ex-
plicit determination is another matter.
A constructive solution of the Pauli Problem has been
obtained for a restricted set of states in a one-particle Hilbert
space @13#. Imagine that the system is known to be prepared
in a state which is made up of a finite but arbitrarily large
number of energy eigenstates of a harmonic oscillator. With
PHYSICAL REVIEW A APRIL 1996VOLUME 53, NUMBER 4
53
1050-2947/96/53~4!/2078~6!/$10.00 2078 © 1996 The American Physical Society
this additional information one is able to reconstruct the state
from the knowledge of position and momentum distributions
over the real line. Another proposition @14# expresses the
quantum state in terms of expectation values of simple pro-
jection operators, and a quantum optical realization of this
approach seems feasible.
It is the goal of the present paper to investigate the Pauli
problem for a particle in analogy to the method developed in
@3~a!#. To do so, the method to solve the Pauli problem for a
spin is, in Sec. II, briefly reviewed, and then it is reinter-
preted in terms of particle dynamics. In Sec. III the main
result is established: it is possible to enumerate all pure
quantum states compatible with the spatial probability den-
sities at two times t and t
8
, separated by a short-time inter-
val, Dt. Finally, in Sec. IV it is shown how to distinguish
between these Pauli partners through appropriate additional
measurements.
II. REINTERPRETATION OF THE PAULI PROBLEM
FOR A SPIN
In this section a solution of the Pauli problem for a spin of
length s is briefly reviewed. Then, the method of solution is
reinterpreted in such a way that an analogous treatment of
the Pauli problem for a particle becomes possible. It has been
shown in @3~a!# that the measurement of the intensities of a
spin state
u
x
&
along two neighboring axes of quantization,
z and z
8
, by means of a Stern-Gerlach apparatus is compat-
ible with a discrete set N (s) of states. An additional mea-
surement of the expectation value of one well-defined opera-
tor allows one to discriminate between the elements of the
set N (s).
More specifically, one proceeds as follows. It is assumed
that there is a beam of particles propagating along the x axis,
say, each of which is prepared in one and the same pure spin
state
u
x
&
in the ~2s11!-dimensional Hilbert space H. They
enter a Stern-Gerlach apparatus that defines the axis of quan-
tization to be along the z direction. The associated eigen-
states of the z component of the spin,
$
u
m,z
&
%
, constitute a
basis in Hilbert space, and the quantum number m takes on
all ~half-! integer values between 6s. In a first series of
measurements all intensities
u
x
m
(z)
u
2
[
u
^
m,z
u
x
&
u
2
are deter-
mined with respect to the z basis. The states compatible with
these measurements are located on a 2s-dimensional mani-
fold A(s): the phase of each state with respect to the basis
u
m,z
&
is undetermined, but the overall phase of the state
u
c
&
does not have physical meaning since a state of the sys-
tem is associated with a ray in H. The set of states corre-
sponding to the elements of the manifold A(s) will be de-
noted by
u
x
(
g
)
&
, where the label
g
5 (
g
1
,...,
g
2s
)
parameterizes the manifold A(s). It is assumed that the
phase conventions are chosen in such a way that the actual
state of the system corresponds to
g
5 0:
u
x
(0)
&
[
u
x
&
.
Consider now an infinitesimal coordinate transformation,
U
ˆ
~
e
!
5 exp
F
2
i
e
\
S
ˆ
x
G
5 12
i
e
\
S
ˆ
x
1 O
~
e
2
!
, ~1!
corresponding to a rotation of the Stern-Gerlach apparatus by
e
about the x axis. It defines a new direction of quantization,
z
8
, which is infinitesimally close to z and perpendicular to
the x axis. The measurement of the intensities
u
x
m
(z
8
)
u
2
along z
8
represents 2s more conditions on the possible
states:
u
^
m,z
8
u
x
~
g
!
&
u
2
5
u
x
m
~
z
8
!
u
2
, m52s,2s11,...,1s.
~2!
It can be shown that 2
2s
states on the 2s-dimensional mani-
fold A(s) are compatible with the measurements ~2!; they
represent the discrete set of Pauli partners, N (s). Finally, by
measuring in addition the expectation value of the operator
S
ˆ
x
, one can distinguish between the individual Pauli partners
that lead to identical intensities along the z and z
8
axis.
As it stands, this method to determine a quantum-
mechanical state from measurements cannot be applied di-
rectly to perform the same task for a particle wave-function.
However, one can rephrase this procedure in such a way that
it becomes possible to transfer the method to the particle
problem. Change from the passive view of the rotation to an
active point of view. Then the Stern-Gerlach apparatus de-
fines once and for all an axis of quantization pointing along
z; the measurements, however, are now performed with re-
spect to the states
u
c
&
and U
ˆ
(2
e
)
u
c
&
, having been rotated
by (2
e
) about the x axis relative to the apparatus. Consider
e
as a time parameter, then the operator U
ˆ
of Eq. ~1! can be
conceived as the time evolution operator of a spin in a con-
stant magnetic field B
i
e
x
of appropriate strength, generated
by the Hamiltonian
H
ˆ
52B•m
ˆ
5S
ˆ
x
, ~3!
m
ˆ
being the magnetic moment of the spin. In other words,
the measurement of the intensities along two different spatial
directions, z and z
8
, is equivalent to a measurement of the
intensities associated with the state
u
x
(t)
&
and its time evo-
lution
u
x
(t1 Dt)
&
for Dt5
e
with respect to one fixed direc-
tion, z. This point of view motivates an investigation of the
problem for a particle with Pauli data given by the position
probability densities
u
c
(x,t)
u
2
and
u
c
(x,t1 Dt)
u
2
of a state
u
c
&
at two nearby times.
III. DETERMINATION OF PAULI PARTNERS
FOR A PARTICLE
A particle with mass m is assumed to move in a one-
dimensional potential V(x
ˆ
), described by the Schro
¨
dinger
equation
i\
d
dt
u
c
~
t
!
&
5H
ˆ
u
c
~
t
!
&
[
S
p
ˆ
2
2m
1V
~
x
ˆ
!
D
u
c
~
t
!
&
. ~4!
Suppose that the system is prepared at time t in a normalized
pure state
u
c
(t)
&
with finite energy,
^
c
u
H
ˆ
u
c
&
, `; a fortiori,
the expectation value of the kinetic energy T
ˆ
^
c
u
T
ˆ
u
c
&
5
^
c
u
p
ˆ
2
/2m
u
c
&
,`, ~5!
is finite.
The Hamiltonian H
ˆ
defined in Eq. ~4! generates the time
evolution of the particle state
u
c
&
just as the Hamiltonian
53
2079HOW TO DETERMINE A QUANTUM STATE BY . . .
~3! did for the spin. By analogy, in this section the Pauli
partners for a particle will be determined that are compatible
with the measurement of the probability distribution of posi-
tion at time t and at a short time later, t1 Dt; in the mean-
time, the state is assumed to evolve according to ~4!.
In the position representation one can decompose the
wave function as
^
x
u
c
~
t
!
&
5
u
c
~
x,t
!
u
e
i
f
~
x,t
!
, ~6!
with positive modulus
u
c
(x,t)
u
and a real phase
f
(x,t),
which is assumed to be a continuous function of x. Discon-
tinuities of the phase
f
(x,t) at points where the amplitude
u
c
(x,t)
u
of the wave function vanishes ~i.e., at its nodes!,
however, are not excluded.
Under these assumptions the measurement of the prob-
ability distributions
u
c
(x,t)
u
2
and
u
c
(x,t1 Dt)
u
2
at two
times t and t1 Dt with Dt! 1 will be seen to be compatible
with an M -dimensional manifold M of states, M being the
number of nodes of the wave function under consideration.
The set of Pauli partners turns out to be isomorphic to an
M -dimensional torus, T
M
.
Suppose that at time t the probability distribution
u
c
(x,t)
u
2
has been determined experimentally. All states
compatible with this distribution can be written
c
„x,t;
j
~
x,t
!
…[
c
~
x,t
!
e
i
j
~
x,t
!
5
u
c
~
x,t
!
u
e
i
@
f
~
x,t
!
1
j
~
x,t
!
#
,
~7!
j
(x,t) being a real function. For later convenience, the func-
tion
j
(x,t) is defined as the deviation from the true ~but yet
unknown! phase
f
(x,t); the strategy will be to eliminate the
‘‘freedom’’ in the choice of the function
j
by constraints
imposed by additional measurements. The relation between
the amplitudes of the actual state
u
c
&
at times t and
t
8
5 t1 Dt follows from Schro
¨
dinger’s equation, Eq. ~4!,
c
~
x,t1 Dt
!
5
c
~
x,t
!
1
]
c
~
x,t
!
]
t
Dt1 O
~
Dt
2
!
5
c
~
x,t
!
2
i
\
H
ˆ
c
~
x,t
!
Dt1 O
~
Dt
2
!
. ~8!
Multiplying this equation with the corresponding expansion
of
c
*
(x,t1 Dt) leads to
u
c
t
8
u
2
5
u
c
t
u
2
1
i
\
~
c
t
H
ˆ
c
t
*
2
c
t
*
H
ˆ
c
t
!
Dt1 O
~
Dt
2
!
5
u
c
t
u
2
2
i\
2m
~
c
t
]
xx
c
t
*
2
c
t
*
]
xx
c
t
!
Dt1O
~
Dt
2
!
,
~9!
showing that, to first order in Dt, the change of the modulus
of the wave function does not depend on the potential
V(x). From now on, the time argument of the wave function
will be given as a lower index, and the dependence on x is
suppressed; furthermore,
]
x
[
]
/
]
x, etc. The corresponding
relation for the states given in Eq. ~7! reads
u
c
t
8
~
j
!
u
2
5
u
c
t
~
j
!
u
2
2
i\
2m
„
c
t
~
j
!
]
xx
c
t
*
~
j
!
2
c
t
*
~
j
!
]
xx
c
t
~
j
!
…Dt1O
~
Dt
2
!
. ~10!
The second term on the right-hand-side of this equation can
be written as
2
i\
2m
„
c
t
]
xx
c
t
*
2
c
t
*
]
xx
c
t
22i
~
c
t
]
x
c
t
*
1
c
t
*
]
x
c
t
!
]
x
j
22i
u
c
t
u
2
]
xx
j
…, ~11!
with the explicit form of the second derivative of the wave
function
c
t
(
j
) given by
]
xx
c
t
~
j
!
5„
]
xx
c
t
12i
]
x
c
t
]
x
j
1i
c
t
]
xx
j
2
c
t
~
]
x
j
!
2
…e
i
j
.
~12!
The assumption of identical probability distributions of po-
sition for the states
u
c
&
and
u
c
(
j
)
&
at both times t and t
8
requires the expressions in Eqs. ~9! and ~10! to be equal,
which by using Eq. ~11! implies
~
c
t
]
x
c
t
*
1
c
t
*
]
x
c
t
!
]
x
j
1
u
c
t
u
2
]
xx
j
50, ~13!
a condition that also can be written as
]
x
~
u
c
t
u
2
]
x
j
!
5 0; ~14!
this equation will be referred to as the phase equation. Every
solution
j
(x) of this equation defines a wave function com-
patible with the observed probability distributions at times
t and t
8
, to first-order in Dt. Formally, this equation is iden-
tical to the amplitude transport equation known from semi-
classical quantum mechanics @15#. In correspondence with
the result for the spin system only information about points
close to x is required: only first- and second-order derivatives
occur, to be compared with the occurrence of at most,
second-order differences in the corresponding equation for
the spin @Eq. ~19! of 3~a!#.
It will be shown in the sequel that the nodal structure of
the wave function determines the manifold of solutions of
the phase equation. Strictly speaking, there is a number of
phase equations with solutions
j
(x): it is not necessary to
smoothly continue the functions
j
(x) on the left and on the
right of a zero of the amplitude, since the phase of the wave
function is undetermined at its nodes. Therefore, it is reason-
able to consider the solutions of Eq. ~14! separately in each
‘‘compartment,’’ defined as a region between two zeros of
the amplitude
u
c
t
(x)
u
2
. Suppose that there are M nodes,
apart from those at x56`. Label the zeros of the amplitude
u
c
t
(x)
u
2
from the left to the right by x
2
,x
1
,...,x
M
,x
1
,
starting with 0 at x
2
52` and ending at M 1 1at
x
1
51`. The
m
th compartment C
m
is defined as that one
on the right of the zero x
m
; the compartment extending to
2 ` will be referred to as C
0
. A wave function with M
nodes defines M1 1 compartments; in particular, for a state
without a node there is just one single compartment, C
0
.
The general solution of the phase equation in compart-
ment C
m
,
m
5 0,1, ...,M is given by
2080 53
STEFAN WEIGERT
j
m
~
x
!
5
a
m
E
x
m
0
x
dy
u
c
t
~
y
!
u
2
1
b
m
, xPC
m
,
a
m
,
b
m
PR,
~15!
where x
m
0
is an arbitrary but fixed point in compartment
C
m
. Since the denominator approaches the value zero at the
boundaries of the compartment C
m
, nonzero values
a
m
would imply that the solutions
j
(x) go to infinity at the
nodes of the wave function. Such a behavior of the phase,
however, is not compatible with the assumption of Eq. ~5!,as
will be shown momentarily: a finite expectation value of the
kinetic energy, T
ˆ
, can be ensured only by all constants
a
m
being equal to zero.
First, the behavior of the integral in Eq. ~15! stemming
from points close to a node is calculated approximately.
Then, the expectation value of the kinetic energy will be
shown to diverge whenever
a
m
is different from zero. Con-
sider, for definiteness, the left endpoint x
m
of the region
C
m
; the following argument can be repeated analogously for
right endpoints. The expansion of the amplitude
P(x)[
u
c
t
(x)
u
in the neighborhood to the right of the point
x
m
yields
c
t
~
x
!
5
$
]
x
P
~
x
m
!
~
x2 x
m
!
1 O „
~
x2 x
m
!
2
…
%
e
i
f
~
x
!
, ~16!
using P(x
m
)5 0, and the term
]
x
P(x) in this expansion is
assumed to be different from zero. It turns out that its van-
ishing, corresponding to a nongeneric coincidence of a node
and and an extremum of the modulus, would make things
even worse ~cf. below!. Therefore, the probability density
near the point x
m
is approximately given by
u
c
t
~
x
!
u
2
5 „
]
x
P
~
x
m
!
~
x2 x
m
!
…
2
1 O „
~
x2 x
m
!
3
…. ~17!
Consider now a wave function
c
t
0
(x) with phase
j
m
(x),
obtained from using ~15! in ~17!; the expectation value of the
kinetic energy in a small interval D(
d
)5 (x
m
1
d
,x
m
1
d
0
),
with 0,
d
,
d
0
! 1 and
d
0
fixed, is given by
^
c
~
j
!
u
T
ˆ
u
c
~
j
!
&
D
~
d
!
5
2 \
2
2m
E
x
m
1
d
x
m
1
d
0
dx
c
t
*
e
2i
j
m
]
xx
~
c
t
e
i
j
m
!
5
2\
2
2m
E
x
m
1
d
x
m
1
d
0
dx„
c
t
*
]
xx
c
t
2
u
c
t
u
2
~
]
x
j
m
!
2
~18!
1 i
u
c
t
u
2
]
xx
j
m
12i
c
t
*
]
x
c
t
]
x
j
m
…. ~19!
In the limit
d
→ 0 the main contribution is due to the second
term on the right-hand side being proportional to
a
m
2
E
x
m
1
d
x
m
1
d
0
dx
~
x2x
m
!
2
;
2
a
m
2
x2x
m
U
x
m
1
d
x
m
1
d
0
5
a
m
2
S
1
d
2
1
d
0
D
→` if
d
→ 0. ~20!
The origin of the divergence is obvious: if the phase
j
m
goes
to infinity for
d
→ 0, then the wave function
c
(x) acquires a
more and more rapidly oscillating phase factor,
j
m
(x). Such
oscillations correspond to large values of the energy. Conse-
quently, the constants
a
m
are necessarily equal to zero for all
compartments C
m
, and the resulting dependence of the
phase
j
on x is strongly limited: it has to be constant within
each compartment C
m
:
j
~
x
!
5
b
~
x
!
[
(
m
5 0
M
b
m
x
m
~
x
!
,
b
5
~
b
0
,
b
1
,...,
b
M
!
,
b
m
P
@
0,2
p
!
, ~21!
where the characteristic function of the compartment C
m
has
been introduced:
x
m
(x)5 1ifxPC
m
, and 0 else. A simi-
lar argument also applies to compartment C
0
or, equiva-
lently, to a state without any zero, such as the ground state of
the potential V(x); in this case the point x
2
52` is consid-
ered to be the node. In retrospect, it is now obvious that the
situation does not change qualitatively if the first nonzero
derivative of the expansion of the wave function
u
c
t
u
2
in Eq.
~17! were of higher order: the oscillations would become
even stronger.
Thus, after having measured the probability densities of
position at times t and t
8
, one is able to define the underlying
state up to (M 1 1) constant phases in the (M 11) compart-
ments. The absolute value of the phase of the wave function
is arbitrary so that there remains an M-dimensional manifold
M of states that is compatible with the experimentally de-
termined data. Since the values
b
m
are restricted to the inter-
val
@
0,2
p
) the manifold of Pauli partners is seen to to coin-
cide with an M -dimensional torus: M5 T
M
.
IV. TELLING PAULI PARTNERS APART
ON THE TORUS T
M
The measurement of the probability distributions
u
c
(x,t)
u
2
and
u
c
(x,t1 Dt)
u
2
is sufficient to determine the
wave function
c
(x,t)uptoMrelative phases, where M is
the number of nodes of the wave function under consider-
ation. This set of states is conveniently described in the form
c
~
x,
b
!
5
c
~
x
!
e
i
b
~
x
!
5
(
m
5 0
M
c
m
~
x
!
e
i
b
m
, ~22!
where
c
m
(x)5
x
m
(x)
c
(x) is a function identical to
c
(x)in
the compartment C
m
and equal to zero elsewhere. Note that
both, modulus
u
c
m
(x)
u
and phase
f
m
(x), have already been
determined, at least implicitly: for the time being, only the
(M 1 1) real numbers
b
remain unknown @or, equivalently,
the piecewise constant function
b
(x)in~21!#. Thus, the mea-
surement of M appropriate expectation values is expected to
allow one to single out the actual state of the system.
Measuring quantities that refer to one single point in con-
figuration space only, i.e., measuring local operators, will not
provide the required information, since all unknown phases
drop out immediately. Being the generator of spatial transla-
tions, the momentum operator p
ˆ
will be involved in any
nonlocal quantity, in one way or another. For example, imag-
ine shifting the wave function
c
(x,
b
)bya~finite! amount
Dx and consider the scalar product of the original state and
the shifted one,
c
(x1 Dx,
b
). Close to each node there will
53
2081HOW TO DETERMINE A QUANTUM STATE BY . . .