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, 2

57 (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

Niels Bohr

1885 - 1962

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.

Period 2 – wave mechanics

Heisenberg – matrix mechanics. De Broglie – associating a wave with every ma

terial particle. Schrödinger – equation of wave. Born – statistical interpretation of wavefunction.

Mathematical equivalence of Schrödinger’s wa

ve 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. Griffith

s (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 self-adjoint 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 eigenvect

ors |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 tim

e 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

Stochastic quantization method

∑path e iS/ℏ

where S=∫Ldt is action of a path

Classical particles subject to

random diffusion

Formalism of quantum

mechanics

Predictions for experimental results

Formalism is not the whole story

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: If the nucleus decays, the Geiger counter will sense it and trigger the release of 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.

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 mechanics when going from subatomic to macroscopic systems.

(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’s friendWigner

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

27

The total system

Measuring apparatus Ʀ

environment

Ɛ of 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。

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 EPR-type 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.)

Measurement does not have a clear definition!

It relates quantum and classical worlds, so can not be defined in either

of the two.

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

Physical state can be understood as

relative to

(1) the state of the other part of the composite

system

(2) a particular “branch” of a constantly

“splitting” universe

(3) a particular “mind” in the set of minds of the conscious observer

Everett’s original

proposal

Many-worlds interpretation

Many-minds interpretation

Decoherence effectively playsthe role of splitting 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 matrixto 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.

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.

A history is expressed as Hα={P1(t1),P2(t2), …., Pn(tn)},

where Pi(t)=U†(t0,t)Pi(t0)U(t0,t).

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 so-called 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?

2. Any system is subjected to the influence of environment.

1. Its relevance to measurement problem and interpretation of quantum mechanics. (Could our (classical) experience be explained by quantum mechanics? to what extent?)

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.

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.

environment-induced decoherence

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!