Transcript QM-interpretation

量子力学诠释与测量问题
王文阁
近代物理系, USTC
Outline
I. 几点历史评论
 II. 量子力学形式体系
 III. 纠缠及几个悖论
 VI. 测量问题
 V. 关于量子力学诠释的几个学派

References:





M.Schlosshauer, Rev. Mod. Phys. 76, 1267
(2005).
W.H. Zurek, Rev. Mod. Phys. 75, 715 (2003).
A. Bassi and G.C. Ghirardi, Phys. Rep. 379,
257 (2003).
F.Laloë, Am. J. Phys. 69, 655 (2001).
R. Omnès, Rev. Mod. Phys. 64, 339 (1992).
I. Some historical remarks

About 30 years ago, probably as a result of the
famous discussions between Bohr, Einstein,
Schrödinger, Heisenberg, Pauli, de Broglie, and others,
the majority of physicists thought that the so-called
“Copenhagen interpretation” is the only sensible
attitude for good scientist.

Nowadays, the attitude of physicists is much more
moderate for several reasons: (1) More consist
interpretations have been found. (2) The discoveries
and ideas of Bell. (3) Advances in experimental
techniques makes it possible for fine control of
quantum systems.
History of fundamental quantum
concepts – three periods. Period 1.
Planck – finite grains of energy in
emitting and absorbing radiation, and the
constant h bearing his name.
 Einstein – notion of quantum of light
(photon as named much later).
 Bohr – quantized, permitted orbits and
quantum jumps for atoms.

Max Planck
Planck, Max (1858-1947)
German physicist who formulated an equation
describing the blackbody spectrum in 1900.
Wien and Rayleigh had also developed
equations, but Wien's only worked at high
frequencies, and Rayleigh's only worked at
low frequencies. Planck's spectrum was
obtained by postulating that energy was
directly proportional to frequency (E=hν).
Planck believed that this quantization applied
only to the absorption and emission of energy
by matter, not to electromagnetic waves
themselves. However, it turned out to be much
more general than he could have imagined.
Albert Einstein
March 1905
Einstein sent to the Annalen der Physik a
paper with a new understanding of the
structure of light. He argued that light can act
as though it consists of discrete, independent
particles of energy, in some ways like the
particles of a gas. His revolutionary proposal
seemed to contradict the universally accepted
theory that light consists of smoothly
oscillating electromagnetic waves. But
Einstein showed that light quanta, as he
called the particles of energy, could help to
explain phenomena being studied by
experimental physicists.
Niels Bohr
In 1913 Bohr published a theory about the
structure of the atom, by combining Planck’s
idea of quantized energy and Rutherford's
model of atom. Bohr proposed that electrons
travel only in certain stationary orbits. He
suggested that the outer orbits could hold more
electrons than the inner ones, and that these
outer orbits determine the atom's chemical
properties. Bohr also described the way atoms
emit radiation by suggesting that when an
electron jumps from an outer orbit to an inner
one, it emits light.
Niels Bohr
1885 - 1962
Period 2 – wave mechanics





Heisenberg – matrix mechanics.
De Broglie – associating a wave with every
material particle.
Schrödinger – equation of wave.
Born – statistical interpretation of wavefunction.
Mathematical equivalence of Schrödinger’s
wave mechanics and Heisenberg’s matrix
mechanics, and Dirac’s formal expression.
Werner Karl Heisenberg
Werner Karl Heisenberg (December
5, 1901 – February 1, 1976)
He invented matrix mechanics, the first
formalization of quantum mechanics in 1925.
His uncertainty principle, discovered in 1927,
states that the simultaneous determination of
two paired quantities, for example the position
and momentum of a particle, has an
unavoidable uncertainty. Together with Bohr,
he formulated the Copenhagen interpretation
of quantum mechanics.
Louis de Broglie
Louis de Broglie (August 15, 1892–March 19, 1987),
He received his first degree in history.
His 1924 doctoral thesis introduced his theory of
electron waves. This included the wave-particle
duality theory of matter, based on the work of
Einstein and Planck. This research culminated in
the de Broglie hypothesis stating that any moving
particle or object had an associated wave. Louis
de Broglie thus created a new field in physics, the
wave mechanics, uniting the physics of light and
matter.
Erwin Rudolf Josef Alexander
Schrödinger
Erwin Rudolf Josef Alexander Schrödinger (August
12, 1887 – January 4, 1961), an Austrian physicist
In January 1926, Schrödinger published in the
Annalen der Physik a paper on wave mechanics
and what is now known as the Schrödinger
equation. In this paper he gave a "derivation" of
the wave equation for time independent systems,
and showed that it gave the correct energy
eigenvalues for the hydrogen-like atom. This
paper has been universally celebrated as one of
the most important achievements of the twentieth
century, and created a revolution in quantum
mechanics.
Max Born
Max Born (December 11, 1882 - January 5,
1970) was a mathematician and physicist.
He formulated the now-standard interpretation of
the probability density function for ψ*ψ in the
Schrödinger equation of quantum mechanics.
Debating on the interpretation of
quantum mechanics
Beginning in 1925 a bold new quantum theory
emerged, the creation of a whole generation of
theoretical physicists from many nations. Soon
scientists were vigorously debating how to
interpret the new quantum mechanics. Einstein
took an active part in these discussions.
Heisenberg, Bohr, and other creators of the
theory insisted that it left no meaningful way
open to discuss certain details of an atom's
behavior. For example, one could never predict
the precise moment when an atom would emit
a quantum of light. Einstein could not accept
this lack of certainty; and he raised one
objection after another. At the Solvay
Conferences of 1927 and 1930 the debate
between Bohr and Einstein went on day and
night, neither man conceding defeat.
Period 3 – interpretations






(1) Copenhagen (orthogonal) interpretations,
1920s-1930s.
(2) Additional-variable interpretations. de
Broglie (1926), Bohm (1952).
(3) Relative-state interpretations. Everett
(1957). (many-worlds interpretation).
(4) Modal interpretations. van Fraassen (1973).
(5) Consistent-histories interpretations.
Griffiths (1984).
(6) Physical collapse models. Pearle (1976).
II. Formalism of quantum mechanics
– Axiom I (ref.BG03)

1. Every physical system S is associated to a
Hilbert space H; the physical states of S are
represented by normalized vectors (called
“statevectors”) |ψ> of H. Physical observables
O of the system are represented by selfadjoint operators in H; the possible outcomes
of a measurement of O are given by its
eigenvalues on,

O|on> = on |on>.
 The eigenvalues on are real and the
eigenvectors |on> form a complete
orthonormal set in the Hilbert space H.
Axiom II

2. To determine the state |ψ(t0)> of the system
S at a given initial time t0, a complete set of
commuting observables for S is measured: the
initial statevector is then the unique common
eigenstate of such observables. Its
subsequent time evolution is governed by the
Schrödinger equation

iℏd|ψ(t)>/dt = H|ψ(t)>,
 which uniquely determines the state at any
time once one knows it at the initial time. The
operator H is the Hamiltonian of the system S.
Axioms III and VI

3. The probability of getting, in a measurement
at time t, the eigenvalue on in a measurement
of the observable O is given by
P(on)=|<on|ψ(t)>|2, where |ψ(t)> is the state of
the system at the time in which the
measurement is performed.
 4. The effect of a measurement on the system
S is to drastically change its statevector from
|ψ(t)> to |on>: |ψ(t)> (before measurement) →
|on> (after measurement). This is the famous
postulate of wavepacket reduction (or collapse
of state vector).
Two other quantization methods
Feynman’s path integral
quantization method
∑path e iS/ℏ
where S=∫Ldt is
action of a path
Stochastic quantization
method
Classical particles
subject to
random diffusion
Formalism is not the whole story
Formalism of
quantum
mechanics
Predictions for
experimental results
What is the meaning
of statevector?
Are statevectors the
ultimate representation
of systems?
Is Schrödinger
evolution universal?
Why measurement
processes are
so special?
III. Paradoxes and entanglement –
(1). Schrödinger cat
Schrödinger's cat is a
seemingly paradoxical
thought experiment devised
by Erwin Schrödinger that
attempts to illustrate a
difficulty met in an early
interpretation of quantum
Schrödinger's Cat: If the nucleus
mechanics when going from
decays, the Geiger counter will
subatomic to macroscopic
sense it and trigger the release of
systems.
the gas. In one hour, there is a
50% chance that the nucleus will
decay, and therefore that the gas
will be released and kill the cat.
(2) Wigner’s friend
Wigner's friend is a thought experiment proposed by the physicist
Eugene Wigner; it is an extension of the Schrödinger's cat
experiment designed as a point of departure for discussing the
mind-body problem as viewed by the Copenhagen interpretation of
quantum mechanics.
Essentially, the Wigner's friend experiment asks the question: at
what stage does a "measurement" take place?
It posits a friend of Wigner who performs the Schrödinger's cat
experiment while Wigner is out of the room. Only when Wigner
comes into the room does he himself know the result of the
experiment: until this point, was the state of the system a
superposition of "dead cat/sad friend" and "alive cat/happy
friend," or was it determined at some previous point?
An illustration of Wigner’s friend
Wigner
Wigner’s
friend
Room
(3) Von Neumann’s infinite regress
Consider, for example, Stern-Gerlach spin analyzer.
What we may have, if we have an series (to infinity)
of such analyzers?
How could we
have a definite
experimental
result?
VI. 测量问题

测量问题：薛定谔演化与测量的确定性结果之间的关系。
协调？
测量仪器给出确定的输出
薛定谔演化给出
不同可能结果的
叠加态
environment
The total system
Ɛ of Ʀ
Measuring
apparatus Ʀ
最终目的：建立一个原则上可以对测量仪器进行分析的量子理论。
学术——理论原因：探讨建立一个能够统一描述世界的物理理论的
可能性。
现实——实验原因：使我们有能力分析具有介观、甚至微观大小（
尺度）的测量仪器。
27
Von Neumann’s ideal measurement
scheme
Formulate the problem within the framework of Schrödinger
equation, with S indicating system and A for measurement
apparatus,
Premeasurement
This dynamical process is often referred to as a
premeasurement process.
To complete the description of a measurement, one
needs to solve the following two problems:
(1)The problem of definite outcome.
(2)The problem of preferred basis.
An illustration of the problem of preferred basis.
Consider two spin half particles. The EPRtype entangled state of the system has two
equivalent expressions. Then, which one of
|z> and |x> should be the |sn> state in von
Neumann’s measurement scheme?
V. Six big families of interpretation of
quantum mechanics
There are many interpretations of
quantum mechanics, to explain the
meaning of statevectors and the
measurement problem.
 Most of them belong to six big families of
interpretations discussed in literature.

1. Copenhagen (orthodox) interpretations

Wave packet reduction in measurement. That
is, every measurement induces a
discontinuous change of the statevector of the
system.
 The necessity of classical concepts in order to
describe quantum phenomena, including
measurement. (Classicality is not to be derived from
quantum mechanics.)

There exists a border (“Heisenberg cut”)
between the quantum and the classical worlds.
(Measuring devices and observers are on the classical
side.)
It relates quantum and
classical worlds, so can
not be defined in either
of the two.
Measurement does not
have a clear definition!
2. Additional-variable interpretations
In 1926, de Broglie found that, writing ψ=ReiS,
Schrödinger equation can be written as two equations, a
continuity equation and a Hamilton-Jacobi equation. As
a result, particles can be regarded as moving under a
quantum potential U, in addition to classical potential,
Based on this observation, he proposed his pilot
wave theory. However, after a discussion with Pauli,
de Broglie abandoned his interpretation.
Bohmian mechanics
In 1952, Bohm proposed his version of additional
variables interpretation. Consider Schrödinger equation,
Using Qk to represent the position of the k-th particle,
then,
3. Everett’s relative-state interpretations.
Everett (1957) proposed a “relative state
interpretation”. - In its various forms, it is
sometimes called “many-worlds interpretation”,
or “many-minds interpretations”.
 The central idea of Everett’s proposal is to
assume (i) a statevector for the entire universe
which obeys Schrödinger equation, (ii) all
terms in the superposition of the total state
actually correspond to physical states, at the
completion of measurements.

Basically, what Everett observed is related to
entanglement.
For example, let us consider the entangled state in
Neumann’s ideal measurement scheme,
In Everett’s interpretation, the state |sn> on the right
hand side is meaningful only with respect to |an>.
Various relative-state interpretations
(1) the state of the other
part of the composite
system
Physical state can
be understood as
relative to
(2) a particular “branch”
of a constantly
“splitting” universe
(3) a particular “mind” in the
set of minds of the
conscious observer
Decoherence effectively plays
the role of splitting
Everett’s
original
proposal
Many-worlds
interpretation
Many-minds
interpretation
More modern
viewpoint
4. Modal interpretations.

The first type of modal interpretation was
proposed by van Fraassen (1973). It proposes
to take only empirical adequacy, but not
necessarily “truth” as the goal of science.
 It allows for the assignment of definite
measurement outcomes even if the system is
not in an eigenstate of the observable
representing the measurement.
 Then, unitary evolution may be preserved, to
account for definite measurement results.
Modal interpretations
Their general
goal is
to specify rules of assigning
properties of the density matrix
to physical quantities measured
in experiments.
5.Consistent-histories interpretations




The consistent-(or decoherent-)histories approach was
first introduced by Griffiths (1984) and further
developed by many other authors.
The approach was originally motivated by quantum
cosmology in which the system is a closed system
without external observer.
The basic idea is to study quantum histories and
probabilities of the histories.
Each history is a sequence events represented by a
set of time-ordered projection operators (in
Heisenberg picture).
Consistent-histories interpretations
It gives a logical framework that allows the
discussion of the evolution of a closed quantum
system, without reference to measurement.
A history is expressed as Hα={P1(t1),P2(t2), …., Pn(tn)},
where Pi(t)=U†(t0,t)Pi(t0)U(t0,t).
The usual sum rule for calculating probabilities
requires a consistency condition for two possible
histories.
This condition is necessary, but not sufficient to fix
possible histories.
Framework: a set of basis, from which
the projection operators (events) can be
constructed.
 Incompatible frameworks are allowed.
 Single framework rule: one is allowed to
use one framework only when explaining
the theory. (Otherwise, inconsistency will
appear.)

6. Physical collapse models

The first proposal for theories of this type were
made by Pearle (1976). An important
breakthrough in this direction was the socalled quantum mechanics with spontaneous
localization, proposed by Ghirardi, Rimini, and
Weber (1985).
 The basic idea of such models is to introduce
modification to Schrödinger equation, to
achieve a physical mechanism for wave
packet reduction. Or to include Schrödinger
evolution and wavepacket reduction in a
unified mathematical framework.
quantum mechanics with spontaneous localization
Quantum mechanics with spontaneous localization
intends to supply answers to the following two
problems:
(1)The preferred-basis problem. Which are the states
to which the dynamical reduction process leads?
(2) The system-dependence problem. How can the
coherence-suppressing process become more and
more effective, when going from microscopic to
macroscopic systems?
Assumptions in quantum mechanics with spontaneous
localization
Its further development is the so-called continuous
spontaneous localization model, which is a dynamical
reduction model.
Thank you!
VI. Decoherence program
Decoherence due to environment is one of the
most impressive progresses achieved in
theoretical physical in the past three decades
Why the theory of decoherence is of interest?
1. Its relevance to measurement problem and
interpretation of quantum mechanics. (Could our (classical)
experience be explained by quantum mechanics? to what extent?)
2. Any system is subjected to the influence of
environment.
While the influence of environment on a micro
system may be small in some situations, the
theory shows that the influence on meso and
macro systems is non-negligible from a
quantum mechanical viewpoint.
Reduced density matrix
– an example
Let us consider a system of two entangled subsystems
For an observable Ô that pertains only to system 1,
one can prove
Here ρ1 is reduced density matrix,
Basic idea of decoherence
Let us consider atoms scattering photons. For one
photon,
The reduced density matrix of the atoms is
It is diagonal, if <k-|k+>=0, then, the system of the atoms
is effectively in a mixed state,
Modified von Neumann measurement scheme
To illustrate decoherence program, consider modified
von Neumann measurement scheme, i.e., system +
apparatus + environment.
environment-induced decoherence
The reduced density matrix of system + apparatus is
Many explicit physical models show that the states |en>
of the environment rapidly approach orthogonality, due
to the large number of subsystems composing the
environment, for suitably chosen states |an> which are
called pointer states.
Then, in the basis of the pointer states, the reduced
density matrix is close to diagonal.
The reduced density matrix approaches to a diagonal
matrix
----------- environment-induced
decoherence.
Criterions have been suggested to define preferred
pointer states, e.g.
[Pn,HAE]=0
Pn=|an><an| is the projection operator for a preferred
pointer state of the apparatus and HAE is the
apparatus-environment interaction Hamiltonian.
Predictions of the theory of decoherence (Ref.
Schlosshauer (2005))
Decoherence and interpretations of
quantum mechanics

Let us go back to the six big families of
interpretations of quantum mechanics for
a detailed discussion.
Thank you!