Schrödinger’s Cat Paradox Resolution Using GRW Collapse Model: Von Neumann Measurement Postulate Revisited ()
1. Introduction
As Weinberg recently reminded us [1] , the measurement problem remains a fundamental conundrum. During measurement, the state vector of the microscopic system collapses in a probabilistic way to one of a number of classical states, in a way that is unexplained, and cannot be described by the time-dependent Schrödinger equation [1] . To review the essentials, it is sufficient to consider two-state systems. Suppose a nucleus
whose Hilbert space is spanned by orthonormal states
where
and
is in the superposition state,
(1.1)
A measurement apparatus
which may be microscopic or macroscopic, is designed to distinguish between states
by transitioning at each instant
into state
if it finds that
is in
Assume that the detector is reliable, implying that the
and
are orthonormal at each instant
, i.e.,
and that the measurement interaction does not disturb states
―i.e., the measurement is “ideal”. When
measures
the Schrödinger equation’s unitary time evolution then leads to the “measurement state”
(1.2)
of the composite system
following the measurement.
Standard formalism of continuous quantum measurements [2] [3] [4] [5] leads to a definite but unpredictable measurement outcome, either
or
and that
suddenly “collapses” at instant
into the corresponding state
But unfortunately Equation (1.2) does not appear to resemble such a collapsed state at instant
.
The measurement problem is as follows:
(I) How do we reconcile the canonical collapse model that postulates [2] definite but unpredictable outcomes with the “measurement state”
(II) How do we reconcile the measurement that postulates definite but unpredictable outcomes with the “measurement state”
at each instant
and
(III) How does the outcome become irreversibly recorded in light of the Schrödinger equation’s unitary and, hence, reversible evolution?
This paper deals with only the special case of the measurement problem, known as Schrödinger’s cat paradox (Figure 1). For a good and complete explanation of this paradox one can see Leggett [6] and Hobson [7] .
Figure 1. Schrödinger’s cat adapted to the measurement of position of an alpha particle [8] [9] [10] .
In his famous thought experiment [11] , Schrôdinger (1935) imagined a cat that measures the value of a quantum mechanical observable with its life. Adapted to the measurement of position of an alpha particle, the experiment is this. A cat, a flask of poison, and a radioactive source are placed in a sealed box. If an internal monitor detects radioactivity (i.e. a single atom decaying), the flask is shattered, releasing the poison that kills the cat. The Copenhagen interpretation of quantum mechanics implies that after a while, the cat is simultaneously alive and dead. Yet, when one looks in the box, one sees the cat either alive or dead, not both alive and dead.
This poses the question of when exactly quantum superposition ends and reality collapses into one possibility or the other?
Since Schrödinger’s time, no any interpretations or extensions of quantum mechanics have been proposed which gives clear unambiguous answers to the questions posed by Schrödinger’s cat of how long superpositions last and when (or whether) they collapse.
The canonical interpretations of the experiment
Copenhagen interpretation
The most commonly held interpretation of quantum mechanics is the Copenhagen interpretation [12] . In the Copenhagen interpretation, a system stops being a superposition of states and becomes either one or the other when an observation takes place. This thought experiment makes apparent the fact that the nature of measurement, or observation, is not well-defined in this interpretation. The experiment can be interpreted to mean that while the box is closed, the system simultaneously exists in a superposition of the states “decayed nucleus/dead cat” and “undecayed nucleus/living cat”, and that only when the box is opened and an observation performed does the wave function collapse into one of the two states.
However, one of the main scientists associated with the Copenhagen interpretation, Niels Bohr, never had in mind the observer-induced collapse of the wave function, so that Schrödinger’s cat did not pose any riddle to him. The cat would be either dead or alive long before the box is opened by a conscious observer [13] . Analysis of an actual experiment found that measurement alone (for example by a Geiger counter) is sufficient to collapse a quantum wave function before there is any conscious observation of the measurement [14] . The view that the “observation” is taken when a particle from the nucleus hits the detector can be developed into objective collapse theories. The thought experiment requires an “unconscious observation” by the detector in order for magnification to occur.
Objective collapse theories
According to objective collapse theories, superpositions are destroyed spontaneously (irrespective of external observation) when some objective physical threshold (of time, mass, temperature, irreversibility, etc.) is reached. Thus, the cat would be expected to have settled into a definite state long before the box is opened. This could loosely be phrased as “the cat observes itself”, or “the environment observes the cat”.
Objective collapse theories require a modification of standard quantum mechanics to allow superpositions to be destroyed by the process of time evolution. This process, known as “decoherence”, is among the fastest processes currently known to physics [15] .
Ensemble interpretation
The ensemble interpretation states that superpositions are nothing but subensembles of a larger statistical ensemble. The state vector would not apply to individual cat experiments, but only to the statistics of many similarly prepared cat experiments. Proponents of this interpretation state that this makes the Schrödinger’s cat paradox a trivial matter, or a non-issue. This interpretation serves to discard the idea that a single physical system in quantum mechanics has a mathematical description that corresponds to it in any way.
Remark 1.1. Ensemble interpretation is in a good agreement with a canonical interpretation of the wave function (
-function) in canonical QM-measurement theory. However under rigorous consideration of a dynamics of the Schrödinger’s cat, this interpretation gives unphysical result (see Proposition 3.2. (ii)).
The canonical collapse models
In order to appreciate how canonical collapse models work, and what they are able to achieve, we briefly review the GRW model. Let us consider a system of
particles which, only for the sake of simplicity, we take to be scalar and spinless; the GRW model is defined by the following postulates: (1) The state of the system is represented by a wave function
belonging to the Hilbert space
(2) At random times, the wave function experiences a sudden jump of the form:
(1.3)
where
is the state vector of the whole system at time
immediately prior to the jump process and
is a linear operator which is conventionally chosen equal to:
(1.4)
where
is a new parameter of the model which sets the width of the localization process, and
is the position operator associated to the m-th particle of the system and the random variable
corresponds to the place where the jump occurs. (3) It is assumed that the jumps are distributed in time like a Poissonian process with frequency
this is the second new parameter of the model. (4) Between two consecutive jumps, the state vector evolves according to the standard Schrödinger equation.
The 1-particle master equation of the GRW model takes the form
(1.5)
Here
is the standard quantum Hamiltonian of the particle, and
represents the effect of the spontaneous collapses on the particle’s wave function. In the position representation, this operator becomes:
(1.6)
Another modern approach to stochastic reduction is to describe it using a stochastic nonlinear Schrödinger equation, an elegant simplied example of which is the following one particle case known as Quantum Mechanics with Universal Position Localization [QMUPL]:
(1.7)
Here
is the position operator,
it is its expectation value, and
is a constant, characteristic of the model, which sets the strength of the collapse mechanics, and it is chosen proportional to the mass
of the particle according to the formula:
where
is the nucleon’s mass and
measures the collapse strength. It is easy to see that Equation (1.5) contains both non-linear and stochastic terms, which are necessary to induce the collapse of the wave function. For example let us consider a free particle (
), and a Gaussian state:
(1.8)
It is easy to see that
given by Equation (1.6) is solution of Equation (1.5), where
(1.9)
The CSL model is defined by the following stochastic differential equation in the Fock space:
(1.10)
2. Generalized Gamov Theory of the Alpha Decay via Tunneling Using GRW Collapse Model
By 1928, George Gamow had solved the theory of the alpha decay via tunneling [8] . The alpha particle is trapped in a potential well by the nucleus. Classically, it is forbidden to escape, but according to the (then) newly discovered principles of quantum mechanics, it has a tiny (but non-zero) probability of “tunneling” through the barrier and appearing on the other side to escape the nucleus. Gamow solved a model potential for the nucleus and derived, from first principles, a relationship between the half-life of the decay, and the energy of the emission.
The
-particle has total energy
and is incident on the barrier from the right to left.
Figure 2. The particle has total energy
and is incident on the barrier
from right to left. Adapted from [8] .
The Schrödinger equation in each of regions
and
takes the following form
(2.1)
where
(2.2)
The solutions read [8] :
(2.3)
where
(2.4)
At the boundary
we have the following boundary conditions:
(2.5)
At the boundary
we have the following boundary conditions
(2.6)
From the boundary conditions (2.5)-(2.6) one obtains [8] :
(2.7)
From (2.7) one obtain the conservation law
Let us introduce now a function
where
(2.8)
Assumption 2.1. We assume now that:
(i) at instant
the wave function
experiences a sudden jump of the form
(2.9)
where
is a linear operator which is chosen equal to:
(2.10)
where
Remark 2.1. Note that:
(ii) at instant
the wave function
experiences a sudden jump of the form
(2.11)
where
is a linear operator which is chosen equal to:
(2.12)
Remark 2.2. Note that:
(iii) at instant
the wave function
experiences a sudden jump of the form
(2.13)
where
is a linear operator which is chosen equal to:
. (2.14)
Remark 2.3. Note that we have chosen operators (2.10), (2.12) and (2.14) such that the boundary conditions (2.5), (2.6) are satisfied.
Definition 2.1. Let
be a solution of the Schrödinger Equation (2.1). The stationary Schrödinger Equation (2.1) is a weakly well preserved in region
by collapsed wave function
if there exist an wave function
such that the estimate
(2.15)
where
is satisfied.
Proposition 2.1. The Schrödinger equation in each of regions
is a weakly well preserved by collapsed wave function
and
correspondingly.
Proof. See Appendix B.
Definition 2.2. Let us consider the time-dependent Schrödinger equation:
(2.16)
The time-dependent Schrödinger Equation (2.16) is a weakly well preserved by corresponding to
collapsed wave function
in region
if there exist an wave function
such that the estimate
(2.17)
where
is satisfied.
Definition 2.3. Let
be a function
Let us consider the Probability Current Law
(2.18)
corresponding to Schrödinger Equation (2.16). Probability Current Law (2.18) is a weakly well preserved by corresponding to
collapsed wave function
in region
if there exist an wave function
such that the estimate
(2.19)
where
is satisfied.
Proposition 2.2. Assume that there exists an wave function
such that the estimate (2.17) is satisfied. Then Probability Current Law (2.18) is a weakly well preserved by corresponding to
collapsed wave function
in region
i.e. the estimate (2.19) is satisfied on the wave function
.
3. Schrödinger’s Cat Paradox Resolution
In this section we shall consider the problem of the collapse of the cat state vector on the basis of two different hypotheses:
(A) The canonical postulate of QM is correct in all cases.
(B) The canonical interpretation of the wave function
is correct only when the supports the wave functions
and
essentially overlap. When the wave functions
and
have separated supports (as in the case of the experiment that we are considering in section II) we claim that canonical interpretation of the wave function
is no longer valid for a such cat state (for details see Appendix C).
3.1. Consideration of the Schrödinger’s Cat Paradox Using Canonical Von Neumann Postulate
Let
and
be
(3.1)
In a good approximation we assume now that
(3.2)
and
(3.3)
Remark 3.1. Note that:
(i)
(ii) Feynman propagator of a free
-particle is [9] :
. (3.4)
Therefore from Equations ((3.3), (2.9) and (3.4)) we obtain
(3.5)
where
(3.6)
We assume now that
(3.7)
Oscillatory integral in RHS of Equation (3.5) is calculated now directly using stationary phase approximation. The phase term
given by Equation (3.6) is stationary when
(3.8)
Therefore
(3.9)
and thus stationary point
are
(3.10)
Thus from Equation (3.5) and Equation (3.10) using stationary phase approximation we obtain
(3.11)
where
(3.12)
From Equation (3.10) we obtain
(3.13)
Remark 3.2. From the inequality (3.7) and Equation (3.13) follows that
-particle at each instant
moves quasiclassically from right to left by the law
(3.14)
i.e., estimating the position
at each instant
with final error
gives
with a probability
Remark 3.3. We assume now that a distance between radioactive source and internal monitor which detects a single atom decaying (see Figure 1) is equal to
Proposition 3.1. After
-decay at instant
the collaps:
arises at instant
(3.15)
with a probability
to observe a state
at instant
is
Proof. Note that. In this case Schrödinger’s cat in fact performs the single measurement of
-particle position with accuracy of
at instant
(given by Equation (3.15)) by internal monitor (see Figure 1). The probability of getting a result
with accuracy of
given by
(3.16)
Therefore at instant
the
-particle kills Schrödinger’s cat with a probability
Remark 3.4. Note that. When Schrödinger’s cat has performed this measurement the immediate post measurement state of
-particle (by von Neumann postulate C.4) will end up in the state
(3.17)
From Equation (3.17) one obtains
(3.18)
Therefore the state
again kills Schrödinger’s cat with a probability
Suppose now that a nucleus
whose Hilbert space is spanned by orthonormal states
where
and
is in the superposition state
(3.19)
Remark 3.5. Note that: (i)
(ii) Feynman propagator of
-particle inside region
are [9] :
(3.20)
where
(3.21)
Therefore from Equations ((2.11), (2.12) and (3.20), (3.21)) we obtain
(3.22)
where
Remark 3.6. We assume for simplification now that
(3.23)
Therefore oscillatory integral in RHS of Equation (3.22) is calculated now directly using stationary phase approximation. The phase term
given by Equation (3.21) is stationary when
(3.24)
and thus stationary point
are
(3.25)
Thus from Equation (3.22) and Equation (3.25) using stationary phase approximation we obtain
(3.26)
Therefore from Equation (3.22) and Equation (3.26) we obtain
. (3.27)
Remark 3.7. Note that for each instant
Remark 3.8. Note that, from Equations ((3.11), (3.13), (3.19), (3.22)-(3.27) and (A.13)) by Remark 3.7 we obtain
(3.28)
Proposition 3.2. (i) Suppose that a nucleus
is in the superposition state
(
-particle) given by Equation (3.19). Then the collaps:
arises at instant
(3.29)
with a probability
to observe a state
at instant
is
(ii) Assume now a Schrödinger’s cat has performed the single measurement of
-particle position with accuracy of
at instant
(given by Equation (3.29)) by internal monitor (see Figure 1) and the result
is not observed by Schrödinger’s cat. Then the collaps:
never arises at any instant
and a probability
to observe a state
at instant
is
Proof.(i) Note that for
the marginal density matrix
is diagonal
In this case a Schrödinger’s cat in fact performs the single measurement of
-particle position with accuracy of
at instant
(given by Equation (3.29)) by internal monitor (see Figure 1). The probability of getting a result
at instant
with accuracy of
given by
(3.30)
From Equation (3.30) by Remark 3.7 and Equation (3.13) one obtains
(3.31)
Note that. When Schrödinger’s cat has performed this measurement and the result
is observed, then the immediate post measurement state of
-particle (by von Neumann postulate C.4) will end up in the state
(3.32)
From Equation (3.32) by Equation (3.31) and by Remark 3.7 one obtains
Obviously by Remark 3.4 the state
kills Schrödinger’s cat with a probability
Proof.(ii) The probability of getting a result
at any instant
with accuracy of
by Equation (3.31) and Equation (3.13) given by
Thus standard formalism of continuous quantum measurements [2] [3] [4] [5] leads to a definite but unpredictable measurement outcomes, either
or
and thus
suddenly “collapses” at unpredictable instant
into the corresponding state
3.2. Resolution of the Schrödinger’s Cat Paradox Using Generalized Von Neumann Postulate
Proposition 3.3. Suppose that a nucleus
is in the superposition state given by Equation (3.19). The collaps:
arises at instant
(3.33)
with a probability
to observe a state
at instant
is
Proof. Let us consider now a state
given by Equation (3.19). This state consists of a sum of two wave packets
and
Wave packet
present an
-particle which lives in region
with a pro- bability
(see Figure 2). Wave packet
present an
-particle which lives in region
with a probability
(see Figure 2) and moves from the right to the left. Note that
From Equation (3.28) follows that
-particle at each instant
moves quasiclassically from right to left by the law
(3.34)
From Equation (3.34) one obtains
. (3.35)
Note that, in this case Schrödinger’s cat in fact performs a single measurement of
-particle position with accuracy of
at instant
(given by Equation (3.35)) by internal monitor (see Figure 1). The probability of getting the result
at instant
with accuracy of
by Remark 3.7 and by postulate C.V.2 and by postulate C.IV.3 (see Appendix C) given by (for complete explanation and motivation see [16] )
(3.36)
Note that, when Schrödinger’s cat has performed this measurement and the result
is observed, then the immediate post measurement state of
-particle (by generalized Von Neumann postulate C.V.3) will end up in the state
(3.37)
The state
again kills Schrödinger’s cat with a probability
Thus is the collapsed state of the cat always shows definite and predictable outcomes even if cat also consists of a superposition:
.
Contrary to van Kampen’s [10] and some others’ opinions, “looking” at the outcome changes nothing, beyond informing the observer of what has already happened.
We remain: there are widespread claims that Schrödinger’s cat is not in a definite alive or dead state but is, instead, in a superposition of the two. van Kampen, for example, writes “The whole system is in a superposition of two states: one in which no decay has occurred and one in which it has occurred. Hence, the state of the cat also consists of a superposition:
. The state remains a superposition until an observer looks at the cat” [10] .
4. Conclusions
A new quantum mechanical formalism based on the probability representation of quantum states is proposed (for complete explanation see [17] ). This paper in particular deals with the special case of the measurement problem, known as Schrödinger’s cat paradox. We pointed out that Schrödinger’s cat demands to reconcile Born’s rule. Using new quantum mechanical formalism we find that the collapsed state of the Schrödinger’s cat always shows definite and predictable outcomes even if cat also consists of a superposition (see [16] [17] [18] )
Using new quantum mechanical formalism the EPRB-paradox is considered successfully. We find that the EPRB-paradox can be resolved by nonprincipal and convenient relaxing of the Einstein’s locality principle.
Acknowledgements
We thank the Editor and the referee for their comments.
Appendix A
The time-dependent Schrodinger equation governs the time evolution of a quantum mechanical system:
(A.1)
The average, or expectation, value
of an observable
corresponding to a quantum mechanical operator
is given by:
. (A.2)
Remark A.1. We assume now that: the solution
of the time- dependent Schrödinger Equation (A.1) has a good approximation by a delta function such that
(A.3)
Remark A.2. Note that under conditions given by Equation (A.3) QM-system which governed by Schrödinger Equation (A.1) completely evolve quasiclassically i.e. estimating the position
at each instant
with final error
gives
with a probability
Thus from Equation (A.2) and Equation (A.3) we obtain
(A.4)
Thus under condition given by Equation (A.3) one obtains
(A.5)
Remark A.3. Let
be the solutions of the time-depen- dent Schrödinger Equation (A.1). We assume now that
is a linear superposition such that
(A.6)
Then we obtain
(A.7)
Definition A.1. Let
be a vector-function
(A.8)
where
(A.9)
Definition A.2. Let
be a vector-function
(A.10)
where
(A.11)
Substituting Equation (A.11) into Equation (A.9) gives
(A.12)
Substitution Equation (A.5) into Equation (A.12) gives
(A.13)
Appendix B
The Schrödinger Equation (2.1) in region
has the following form
. (B.1)
From Schrödinger Equation (B.1) it follows
(B.2)
Let
be a function
(B.3)
where
(B.4)
see Equation (2.9). Note that
(B.5)
Therefore substitution (B.2) into LHS of the Schrödinger Equation (B.1) gives
(B.6)
Note that
(B.7)
Therefore from Equation (B.6) and Equations ((2.3) and (2.4)) one obtains
(B.8)
From Equation (B.6) one obtains
(B.9)
From Equation (B.9) and Equations ((2.3), (2.4)) one obtains
(B.10)
and
(B.11)
Appendix C. Generalized Postulates for Continuous Valued Observables
Suppose we have an n-dimensional physical quantum system with continuous observables.
I. Then we claim the following:
C.I. Any given
-dimensional quantum system is identified by a set
where:
(i)
that is some infinite-dimensional complex Hilbert space,
(ii)
that is complete probability space,
(iii)
that is measurable space,
(iv) #Math_345# that is complete space of random variables
such that
(v)
that is one to one correspondence such that
(C.1)
for any
and for any Hermitian operator
where
is
―algebra of the Her- mitian adjoint operators in
and
an commutative subalgebra of
(vi)
is an continuous vector function
which represented the evolution of the quantum system
C.I.2. For any
and for any Hermitian operator
such that
(C.2)
C.I.3. Suppose that the evolution of the quantum system is represented by continuous vector function
Then any process of continuous measurements on measuring observable
for the system in state
one can to describe by an continuous
-valued stochastic processes
given on probability space
and a measurable space
.
Remark C.1. We assume now for short but without loss of generality that
Remark C.2. Let
be random variable
such that
then we denote such random variable by
or simply
for short. The probability density of random variable
we denote by
or simply
for short.
Definition C.1. The classical pure states correspond to vectors
of norm
. Thus the set of all classical pure states corresponds to the unit sphere
in a Hilbert space
.
Definition C.2. The projective Hilbert space
of a complex Hilbert space
is the set of equivalence classes
of vectors
in
, with
for the equivalence relation given by
for some non-zero complex number
The equivalence classes for the relation
are also called rays or projective rays.
Remark C.3. The physical significance of the projective Hilbert space
is that in canonical quantum theory, the states
and
represent the same physical state of the quantum system, for any
. It is conventional to choose a state
from the ray
so that it has unit norm
Remark C.4. In contrast with canonical quantum theory we have used instead contrary to
equivalence relation
a Hilbert space
(see Definition C.7).
Definition C.3. The non-classical pure states correspond to the vectors
of a norm
. Thus the set of all non-classical pure states corresponds to the set
in the Hilbert space
.
Suppose we have an observable
of a quantum system that is found through an exhaustive series of measurements, to have a set
of values
such that
Math_411# Note that in practice any observable
is measured to an accuracy
determined by the measuring device. We represent now by
the idealized state of the system in the limit
for which the observable definitely has the value
II. Then we claim the following:
C.II.1. The states
form a complete set of
-function normalized basis states for the state space
of the system.
That the states
form a complete set of basis states means that any state
of the system can be expressed as:
where supp
and while
-function normalized means that
from which follows
so that
The completeness condition can then be written as
C.II.2. For the system in state
the probability
of obtaining the result
lying in the range
on measuring observable
is given by
(C.3)
for any
Remark C.5. Note that in general case
C.II.3. The observable
is represented by a Hermitian operator
whose eigenvalues are the possible results
of a measurement of
and the associated eigenstates are the states
i.e.
Remark C.6. Note that the spectral decomposition of the operator
is then
. (C.3)
Definition C.4. A connected set in
is a set
that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other.
Definition C.5. The well localized pure states
with a support
correspond to vectors of norm 1 and such that:
is a connected set in
Thus the set of all well localized pure states corresponds to the unit sphere
in the Hilbert space
.
Suppose we have an observable
of a system that is found through an exhaustive series of measurements, to have a continuous range of values
III. Then we claim the following:
C.III.1. For the system in well localized pure state
such that:
(i)
and
(ii)
is a connected set in
, then the probability
of obtaining the result
lying in the range
on measuring observable
is given by
(C.4)
C.III.2.
C.III.3. Let
and
be well localized pure states with
and
correspondingly. Let
and
correspondingly. Assume that
(here
the closure of
is denoted by
) then random variables
and
are independent.
C.III.4. If the system is in well localized pure state
the state
described by a wave function
and the value of observable
is measured once each on many identically prepared system, the average value of all the measurements will be
(C.5)
The completeness condition can then be written as
Completeness means that for any state
it must be the case that
i.e. there must be a non-zero probability to get some result on measuring observable
C.III.5. (von Neumann measurement postulate) Assume that
(i)
and (ii)
is a connected set in
. Then if on performing a measurement of
with an accuracy
the result is obtained
in the range
, then the system will end up in the state
. (C.6)
IV. We claim the following:
C.IV.1 For the system in state
where: (i)
(ii)
is a connected set in
and (iii)
(C.6)
C.IV.2. Assume that the system in state
where (i)
(ii)
is a connected set in
and (iii)
Then if the system is in state
described by a wave function
and the value of observable
is measured once each on many identically prepared system, the average value of all the measurements will be
(C.7)
C.IV.3. The probability
of obtaining the result
lying in the range
on measuring
is
(C.8)
Remark C.7. Note that C.IV.3 immediately follows from C.IV.1 and C.III.2.
C.IV.4. (Generalized von Neumann measurement postulate) If on performing a measurement of observable
with an accuracy
the result is
obtained in the range
, then the system immediately after measurement will end up in the state
(C.9)
C.V.1. Let
where
(i)
(ii)
is a connected sets in
(iii)
and
(iv) #Math_522#
Then if the system is in a state
described by a wave function
and the value of observable
is measured once each on many identically prepared system, the average value of all the measurements will be
(C.10)
C.V. 2. The probability of getting a result
with an accuracy
such that
or
given by
(C.11)
Remark C.8. Note that C.IV.3 immediately follows from C.III.3.
C.V. 3. Assume that the system is initially in the state
If on performing a measurement of
with an accuracy
the result is ob-
tained in the range
, then the state of the system immediately after measurement given by
#Math_536# (C.12)
Definition C.6. Let
be
Definition C.7. Let
be a state
where
and
Let
be an state such that
States
and
is a
-equivalent:
iff
. (C.13)
C.V. For any state
where
and
there exist an state
such that:
Definition C.8. Let
be a state
where
and
Let
be an state such that
States
and
is a
-equivalent (
) iff:
C.VI. For any state
where
and
there exists an state
such that:
Appendix D. The Position Representation: Position Observable of a Particle in One Dimension
The position representation is used in quantum mechanical problems where it is the position of the particle in space that is of primary interest. For this reason, the position representation, or the wave function, is the preferred choice of representation.
D.1. In one dimension, the position
of a particle can range over the values
Thus the Hermitean operator
corresponding to this observable will have eigenstates
and associated eigenvalues
such that:
D.2. As the eigenvalues cover a continuous range of values, the completeness relation will be expressed as an integral:
where
is the wave function associated with the particle at each instant
. Since there is a continuously infinite number of basis states
these states are
-function normalized:
D.3. The operator
itself can be expressed as:
Definition D.1. A connected set is a set
that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other.
D.4. The wave function is, of course, just the components of the state vector
with respect to the position eigenstates as basis vectors. Hence, the wave function is often referred to as being the state of the system in the position representation. The probability amplitude
is just the wave function, written
and is such that
is the probability
of the particle being observed to have a coordinate in the range
to
Definition D.2. Let
be a state
where
and
Let
be an state such that
States
and
is
-equivalent (
) iff
(D.1)
D.5. From postulate C.5 (see Appendix C) follows: for any state
where
and
there exists an state
such that:
Definition D.2. Let
be a state
where
and
Let
be a state such that
States
and
are
-equivalent (
) iff:
D.6. From postulate C.7 (see Appendix C) follows: for any state
where
and
there exists an state
such that:
Definition D.3. The pure state
is a weakly Gaussian in the position representation iff
. (D.2)
where
and
are given functions which depend only on variable
D.7. From statement D.5 it follows: for any state
where
and
is a weakly Gaussian state there exists an weakly Gaussian state
such that:
. (D.3)
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