## Superposition Principle of Quantum Mechanics

Classical superposition principle

Classical waves pass through each other. For a medium with linear restoring forces, any local displacement of the medium will be equal to the displacement represented by the linear combination of the displacements produced by the interfering 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.

Quantum superposition principle

There are some similarities between the classical waves and the quantum mechanical wavefunction. For example, electrons, as in the double-slit experiments, also produce constructive or destructive interference patterns just like the classical waves. But, this is where the similarities end. There is a fundamental difference:

classical waves are real-valued but the quantum mechanical 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 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.

Quantum Superposition is profoundly different from classical superposition

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

What is an example of a “state?” Position. The position of an elementary particle in space is a state of the system. Quantum Mechanics is telling us that the elementary particle exists in all possible positions at the same time. Please read the previous sentence carefully. Quantum Mechanics is NOT saying that the elementary particle can be anywhere at any given moment in time. Quantum Mechanics is telling us that the elementary particle is everywhere – albeit with different probabilities – simultaneously in every moment of time until a measurement is performed, then it appears at a specific location. After the measurement is performed the superposition is broken, the elementary particle no longer exists in all possible positions at the same time. This phenomenon of broken superposition is also known as wavefunction collapse. The wavefunction collapse is very problematic and has been the source of many debates.

Regarding the measurement, keep in mind that “measurement” is meant in the broadest sense. The physical interaction between an electron and a photon is also a type of measurement.

Wavefunction is not directly observable

This is the biggest dilemma in Quantum Mechanics. The Quantum Mechanical wavefunction is NOT directly observable.

What is the ontological status of the wavefunction? Is it physical or is it just a calculational tool? Question remains because wavefunction is not directly observable. This is another reason why we have so many interpretations of Quantum Mechanics.

If the wavefunction is just a calculational tool then we can ignore the philosophical problems posed by the wavefunction collapse. If the wavefunction is physical then we have to explain the physical mechanism of the wavefunction collapse

Copenhagen interpretation of QM ignores the ontological status of the wavefunction

According to the Copenhagen interpretation questions like “is the wavefunction physical?” or “is the wavefunction real?” are meaningless.  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 and the wavefunction collapse – without explaining it. This approach has been very successful and this is why the Copenhagen interpretation is taught in the physics textbooks as the official QM.

MWI (Many Worlds Interpretation) of QM claims that the wavefunction is physical

MWI says that we have to explain the wavefunction collapse. In MWI, the appearance of the wavefunction collapse is explained by the mechanism of quantum decoherence. This requires the assumption that the whole universe is a wavefunction (the universal wavefunction) which is a quantum superposition of infinitely many, non-communicating, parallel universes or quantum worlds. According to MWI the universal wavefunction is physical.

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.

Model of a Model

It is clear to all physicists that physics does not describe the physical reality per se. Physics theories are models of the physical reality.

No theory, mathematical or not, can describe reality because direct perception of reality is not possible. We perceive the physical reality around us through the nervous system. The physical interaction is translated into electrical pulses in our sensory nerves. These pulses are then converted into an information package by the brain and finally the information package is interpreted by the mind. We don’t know what mind is but we know that the incoming information has to be interpreted otherwise there is no perception.

Human mind is interpreting the perception. Physics theories are interpreting the interpretation. If we call the “interpretation” a model, a physics theory is a model of a model. Theory is a mental construct and it is a model of a model. Physics theories are removed from reality by at least 2 levels.

Physics models (theories) may not be close to reality but they still work. They still make accurate predictions about the outcomes of measurements. They also point to new things as in new particles or elements that can exist but have not been observed yet.

Quantum Mechanics may be a calculational tool only – jury is out on this question – but QM works. It gives accurate predictions in terms of probabilities.

Developing a movie from snapshots is extremely difficult in 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.