## QM wavefunction and its many interpretations

Mathematical formalism known as Quantum Mechanics (QM) describes the outcomes of measurements performed with elementary particles when they interact with each other or external fields.

Measurement is the key concept in QM. Measurement is like taking a snapshot of the ever-changing reality. Snapshot is a static picture but the physical reality is dynamic so this static view (measurement) is already different from the reality. When we perceive the snapshot our perception is removed from reality by one more level and finally when we devise a model to describe the measured outcomes the model will be removed from reality by 3 levels. This is the situation with QM.

In classical physics we model the phenomena (dynamic aspects) and hope that measurements (snapshots) conform to the model. Quantum Mechanics works in the opposite direction. Quantum Mechanics was developed by modeling the measurement results (snapshots). The dynamics (time evolution of the quantum system) is an add-on. In other words, it is very difficult to develop a movie from snapshot pictures. This is done in cinema, of course, but in the physics of the microscopic world developing a movie from snapshots is extremely difficult. At each measurement (snapshot) the wavefunction collapses and yields a single state of the system. Stitching together the snapshots belonging to different collapsed states to come up with the dynamical behavior of the system may be totally meaningless.  This is the root cause of the many non-classical aspects of the Quantum Mechanics.

Why so many interpretations of QM?

As explained above QM is removed from reality by 3 levels. This is why there are many interpretations. We try to close the gap between the reality and the model by introducing epistemology and ontology into the mathematical formalism.

Quantum Mechanics does not model the phenomena. Quantum Mechanics models our mental model of the phenomena. Quantum Mechanics is a model of a model.

The poll

Maximilian Schlosshauer, Johannes Kofler, and Anton Zeilinger carried out a poll at a quantum foundations meeting. The most interesting chart from their poll is this one.

Terminology

A word of caution on terminology: quantum mechanical “wavefunction” and “state vector” refer to the same mathematical construct. Some physicists prefer the term “state vector” others prefer “wavefunction.” I like the term “wavefunction.”

Another alternative term for the wavefunction is “amplitude.” As of late 2013, we started hearing more and more about the “amplitude.” Don’t be confused. In the context of QM and Quantum Field Theory (QFT) the “amplitude” is the same as the wavefunction.

What is a wavefunction?

It is a mathematical construct that represents all the information we can know about a microscopic (quantum mechanical) system. The “w” in the wavefunction stands for “whole.” The wavefunction represents the whole system. In Roger Penrose’s words [1]:

“The different parts of the wave[function] cannot be thought of as local disturbances, each carrying on independently of what is happening in a remote region. Wavefunctions have a strongly non-local character; in this sense they are completely holistic entities.”

There is a long history of the wavefunction but I will ignore the historical aspects in this article. I will ignore most of the mathematical aspects as well. I will focus on the conceptual aspects.

There is a mathematical property of the wavefunction that cannot be ignored even in a conceptual discussion. The quantum mechanical wavefunction is a complex valued function. In other words $i=\sqrt{-1}$ and the algebra of complex numbers are essential ingredients of the wavefuntion.

Fundamental property of the wavefunction: quantum superposition

Wavefunction is not really a wave but it has similarities to waves, hence the name “wavefunction.” Classical waves display the property of superposition. What is superposition? Superposition literally means linear addition. Classical waves pass through each other. Any local displacement of the medium will be equal to the displacement represented by the linear combination of the displacements produced by the classical waves. Depending on the phases of the classical waves the displacements they produce may cancel each other or add linearly. Superposition of classical waves produces constructive or destructive interference patterns.

Electrons, as in the double-slit experiments, also produce constructive or destructive interference patterns. This is sometimes known as the wave-particle duality of QM. So, how is the quantum superposition any different from the classic superposition?

Answer is simple but subtle. Classical waves are real-valued but the QM wavefunction is complex-valued. Superposition (linear addition) of complex-valued waves leads to very bizarre effects.

A quantum system (a captured electron, for example) exists in all its theoretically possible states simultaneously; but when measured or observed, it gives a result corresponding to only one of the possible states. The wavefunction is constructed in such a way that all these states are included in the mathematical form of the wavefunction. This is also superposition because we are simply adding the complex-valued mathematical representations of all the possible states linearly.

The key word is “linear.” QM is a linear theory. The whole wavefunction is a linear combination of all the possible states.  Measurement yields only one of the states but the system behaves (evolves) as if it is a whole.

You should pay attention to the statement: a quantum system exists in all its theoretically possible states simultaneously. In QM, at a given moment, an object can exist in many states at once. This sounds crazy! In classical mechanics,  an object can exist in only one state at a given moment in time.

Probabilistic interpretation of the wavefunction proposed by Max Born

Let’s represent the wavefunction by the symbol $\psi$. No assumptions on the mathematical form of it except that it can be complex valued.

Max Born suggested that the probability of finding an elementary particle such as an electron in a given location is $|\psi|^2$. If you want to find the probability of finding this particle in a specified volume then you sum up (integrate) $|\psi|^2$ over that volume. If you sum up (integrate) over the entire space you should get Probability=1. This interpretation forms the core of the Copenhagen interpretation.

Remember, wavefunction is complex-valued. When you square a complex number you can get a negative number. This is why you have to use the “absolute square” to get the probability density. Negative probability is not physically meaningful. Hence the absolute value signs in $|\psi|^2$.

Why do we square it? Why don’t we say $|\psi|$ is the probability density? This is a very good question. The answer is very subtle related to the mathematics of QM but the simplistic answer would be that the mathematics would become very cumbersome with $|\psi|$ as the probability density. I also suspect that “square” operation has deeper meanings. I may write about this in the future.

Ontological status of the wavefunction

Is the wavefunction physical? Does it objectively exist? What is the ontological status of the wavefunction? Is it real or is it just a calculational tool? These are very good questions that are still being asked. Why?

Question remains because wavefunction is not directly observable. This is another reason why we have so many interpretations of QM.

Copenhagen interpretation

According to the Copenhagen interpretation questions like “is the wavefunction physical?” or “is the wavefunction real?” are meaningless.  Copenhagen interpretation ignores the ontology and focuses on the epistemology of the wavefunction.

Copenhagen interpretation says that the reality of the wavefunction is unknowable so why focus on that question. Copenhagen interpretation says that we can only know the measurement outcomes therefore let’s build the entire calculational mechanism using the concepts such as probabilities, observables and operators. This approach has been very successful and this is why the Copenhagen interpretation is taught in the physics textbooks as the official QM.

Copenhagen interpretation was developed by a number of scientists and philosophers during the second quarter of the 20th Century. It is a probabilistic interpretation of nature with the probability of a given outcome of a measurement given by the square of the modulus of the amplitude of the wave function.

Other essential features of the Copenhagen interpretation are Bohr’s complementarity and correspondence principles as well as Heisenberg’s uncertainty principle.

Intrinsic uncertainty

According to QM uncertainty is intrinsic in nature. Intrinsic uncertainty is different from measurement errors. Intrinsic uncertainty refers to a fundamental ontological uncertainy.This means that even when you have the ideal measurement apparatus and perform measurements without disturbing the system, even then not all properties of the system can be measured with precision at the same time.

Complementarity principle

An experiment can show the particle-like properties of matter, or the wave-like properties; in some experiments both of these complementary viewpoints must be invoked to explain the results.

Correspondence principle

The quantum mechanical description of large systems will closely approximate the classical description. This is the correspondence principle.

Collapse of the wavefunction in the Copenhagen interpretation

According to the Copenhagen interpretation a system is completely described by a wavefunction which evolves smoothly in time, except when a measurement is made, at which point it instantaneously collapses to one of the possible states (eigenstate) of the observable that is measured. In the Copenhagen interpretation wavefunction collapse is a fundamental, a priori principle.

First differentiator

The first criterion that differentiates other interpretations of QM from the Copenhagen interpretation is the treatment of the wavefunction collapse. In most other interpretations of QM the wavefunction collapse is rejected as a fundamental, a priori principle.

• In the many-worlds-interpretation (MWI) of QM the “appearance of collapse” is explained by the mechanism of quantum decoherence.
• In the Relational QM interpretation the problem of the collapse is avoided by modifying the notion of state.  Relational QM argues that the “state” describes the correlation between the system and its observers. To one observer at a given point in time, a system may be in a single, “collapsed” eigenstate, while to another observer at the same time, it may be in a superposition of two or more states.
• In the von Neumann – Wigner interpretation the collapse is caused by consciousness (this interpretation is not very popular among physicists)
• In the objective collapse interpretations some amount of nonlinearity is introduced to QM which is a linear theory to explain the wavefunction collapse.

Second differentiator

Another criterion that differentiates other interpretations of QM from the Copenhagen interpretation is the ontological status of the wavefunction. Copenhagen interpretation avoids this question. In MWI and objective collapse interpretations the wavefunction is considered to be real in the physical sense.

MWI

In MWI, the whole universe is a wavefunction (the universal wavefunction) which is a quantum superposition of infinitely many, non-communicating, parallel universes or quantum worlds.

The universal wavefunction is physical and there is no wavefunction collapse.

In MWI, the appearance of the collapse is explained by the mechanism of quantum decoherence. MWI claims to resolve all of the correlation paradoxes of quantum theory, such as the EPR paradox and the “Schrödinger’s cat” parodox.

MWI is also known as the Everett interpretation or the theory of the universal wavefunction. The phrase “many-worlds” is due to Bryce DeWitt who was responsible for the wider popularization of Everett’s theory. DeWitt’s “many-worlds”, Everett’s “Universal Wavefunction” or Everett–Wheeler’s “Relative State Formulation” refer to the same interpretation of QM.

Quantum decoherence

The key concept of MWI is quantum decoherence which occurs when quantum states interact with the environment.

Decoherence interpretations of QM are based on a discussion of “pure states” and “mixed states.” These are very important concepts that I hope to write about in the future.

Information theoretical interpretations of QM

In these interpretations of QM the wavefunction describes an observer’s knowledge of the world, rather than the world itself. This approach has some similarity with Copenhagen interpretation.

Information ontologies interpretations of QM

Information ontologies, such as J. A. Wheeler’s “it from bit“. These approaches have been described as a revival of immaterialism.

Bohmian interpretations

For details of Bohmian Mechanics, see the entry in SEP

http://plato.stanford.edu/entries/qm-bohm/