wave function

If you read my last post on the Thomas Young’s double slit experiment then you may recall the discussion of the collapse of the wave function. If you didn’t read that post, you should, but I will briefly describe the idea of the wave function collapse. The wave function equation is a formula which describes all the information of a particle. The square of the wave function gives you the probability of the location of a particle such as an electron or photon. Quantum mechanics differs from Newtonian mechanics in that it is probabilistic rather than deterministic. This means we can calculate the probability of the location or velocity of an object for example but we don’t know for sure until we make a measurement. In the double slit experiment we don’t know which slit a specific electron or photon travels through. We can use a detector to determine which slit it physically passed through but this measurement has consequences. When we measure or observe the particle we say the wave function collapses and the particle must travel through one slit or the other. Prior to making the observation the particle is in a state of superposition which means it is in all possible locations simultaneously. This idea is the most widely accepted theory regarding quantum behavior and is known as The Copenhagen Interpretation.

Image result for wave function" — Schrodinger’s wave equation

But what if making the measurement doesn’t cause the wave function to collapse. What if, rather, it causes one outcome to occur in our universe and causes all other possible outcomes to occur in a separate universes? This may sound like science fiction but it is considered to be a legitimate theory in quantum mechanics and is called The Many Worlds Theory.

Image result for many worlds theory — A depiction of the may worlds theory at the classical level. Courtesy of britannica.com

Many Worlds Vocabulary

Before we delve farther into this fascinating and controversial interpretation of quantum mechanics let’s review some terminology that will appear in this post. The meaning of superposition with respect to quantum mechanics is that a particle or group of particles such as electrons, photons, even molecule can exist across all possible states simultaneously. Decoherence in quantum mechanics is a way around the wave collapse idea. When quantum features interact with the classical realm then the wave function is no longer a smooth single quantum system. The wave function is said to be no longer coherent, that is smooth and continuous. Another way to think of this is that the particle in question is no longer in a state of superposition. When the decoherence occurs the particle must have a specific state. It is at this point, according to the many worlds theory that the universe branches off and all possible states of the particle occur but in different universes that can never interact with one another.

The World’s Most Famous Cat

Image result for schrodinger's cat" — Schrodinger’s famous though experiment. Courtesy of science.howstuffworks.com

Schrodinger’s cat is a thought experiment developed by Erwin Schrodinger to illustrate what he perceived to be the ridiculousness of The Copenhagen Interpretation of quantum mechanics. In this experiment, a cat is placed in a box with a vile of poison. There is a particle that is undergoing radioactive decay and when the decay is complete it triggers a hammer to smash the vile of poison causing it to be released thus killing the cat. The particle itself is part of the quantum world and so it may or may not decay. A timer is set for one hour, at which time the decay of the particle would be complete. The idea is that once the hour is up, the cat is both alive and dead until we open the box to observe it. It is only by making an observation, opening the box, that we can determine its state. Schrodinger created this thought experiment in an effort to show that the idea of superposition of states is a fundamentally fatal flaw in The Copenhagen Interpretation. We know today that in the quantum world the superposition, does in fact occur, but does it occur at the macroscopic level? Here is a video of from minutephysics that describes Schrodinger’s cat experiment and briefly discusses the implications: https://www.youtube.com/watch?v=IOYyCHGWJq4

What Does The Many Worlds Theory Mean?

If we put science fiction aside for the moment we can look into what this interpretation actually says. As previously stated, The Many Worlds interpretation was developed in response to The Copenhagen Interpretation. What should be made clear is that both interpretations make the same predictions about the eventual outcome of a quantum system. If, for example, you were making quantum predictions about the double slit experiment both interpretations would arrive at the same conclusions. Where they differ is in how you arrive at your conclusion.

The Many Worlds Theory says that any time there is a quantum interaction between particles and the environment, which can be an observer or anything in the macroscopic world. The quantum system experiences decoherence and a single outcome occurs. At this point the theory predicts that “branches” or other universes are formed for every possible alternative of the particle’s superposition of states. It is important to note that once these branches are formed they are completely independent of each other and do not interact with each other. The Many Worlds theory concludes that all branches or universes are equally valid and real but that because quantum is inherently probabilistic some possibilities are more probable. So if we think back to the double slit experiment and measure an electron going through the slit on the left of the grate then at this point a new universe or branch is formed where the electron traveled through the slit on the right hand side.

Where this theory differs from The Copenhagen Interpretation is that in this interpretation there is no wave collapse. In The Many Worlds version, all possible outcomes occur, they just occur in other parallel universes. The elegance of this theory is that it does not require the collapse of the wave function. This is important because the wave function collapse is not derived from Schrodinger’s wave equation and this collapse must be added in order to satisfy The Copenhagen Interpretation. All theories agree that quantum systems follow Schrodinger’s equation when they are not being observed. The Many Worlds Interpretation says that these systems also follow the same equation when they are being observed.

Are There Copies of Us in the Other Universes?

Ok, enough about boring electrons creating and existing in other universes. Are there other versions of me in other branches or universes? According to physicist Sean Carroll at Caltech the answer is yes. “Its all the same. Many Worlds says, look, if an electon can be in a superposition, you can too.” He is quick to point out that a new universe is not created because you made a decision it only occurs when a quantum system interacts with the macroscopic environment.

Who Developed This Theory and Why?

Image result for hugh everett — Hugh Everett III Courtesy of newscientist.com

In 1957 a graduate student by the name of Hugh Everett III published his now famous paper, which was an edited and abbreviated version of an earlier draft, titled “Relative State’s Formulation of Quantum Mechanics”. Everett was dissatisfied with The Copenhagen Interpretation of quantum mechanics specifically with the requirement of the wave collapse. The argument against the wave function is based on the idea that the wave function collapse appears completely arbitrary or random meaning it does not utilize any of the information contained in the wave function to determine which outcome is favored. Secondly, the wave function collapse does not originate from Schrodinger’s equation but rather must be added and some say it violates the Schrodinger equation.

Everett’s ideas were not readily accepted by the scientific community and shortly after publication Hugh Everett went to work for the defense department and never returned to his academic life. Albert Einstein and Erwin Schrodinger also left the field of quantum mechanics to pursue other topics in physics. Einstein had difficulty accepting the probabilistic nature of quantum mechanics. In a letter to colleague Max Born Einstein states “Quantum mechanics … delivers much, but does not really bring us any closer to the secret of the Old One. I, at any rate, am convinced that He does not play dice.” This quote is often abbreviated as “God does not play dice with the universe.” Maybe in another universe Einstein, Schrodinger, and Everett all stayed in the field of quantum mechanics and unraveled more quantum weirdness.

Thomas Young performed his famous “double slit experiment in 1801. The results of this experiment demonstrated the dual nature of light. The term “dual” in the dual nature of light means that light, which is electromagnetic radiation, has both wavelike and particle properties. While the initial experiment was done with light it has since been repeated with electrons, atoms, and even molecules as large as 60 carbon atoms called a Buckyball. All of these experiments show the same dual nature results.

Common terminology

Before we get into the actual experiment and its results lets discuss some of the vocabulary. The first term we should discuss is quantized. If something is quantized then it exists only in discrete continuous amounts. Light is quantized into quanta called photons. Quanta is the fundamental size unit the smallest, indivisible portion of a property. A small packet or chunk of light, for example, which can’t be broken down any further.

Next we need to discuss some terminology about waves. These terms can be used with water waves, light waves, sound waves, basically any type of wave. There are different types of waves, transverse and longitudinal, that both have the same features. The amplitude of a wave is the height of a wave and is directly proportional to the energy of the wave. The higher the wave, the greater the amplitude. Greater amplitude corresponds to a more energetic wave. The top of a wave is called the crest and the low point of a wave is called the trough.

Image result for transverse wave — An image of a transverse wave. Courtesy of study.com

Diffraction is the term used to describe how waves bend around an opening or barrier.

Image result for diffraction of water waves — Note how the pattern changes based on the size of the aperture.
Courtesy of s-cool.co.uk

Wave interference

Waves can interact with other waves or even with itself and this is called interference. There are two basic types of interference. When the peak also called crest of one wave interacts with the crest of another wave then constructive interference occurs. If the trough of one wave interacts with the trough of another wave then the same constructive interference occurs. Constructive interference results in a larger single wave, as if the two parts of the wave that interacted constructed a new single larger wave. If the crest of one wave interacts with the trough of another wave then the waves cancel one another out in a process called destructive interference.

Image result for constructive and destructive interference — Examples of constructive and destructive interference. Courtesy of researchgate.net

The wave function in quantum mechanics is an equation that describes all the characteristics of a particle.

The square of a wave function represents the probability of finding a particle at a given time and place. The square of the wave function becomes significant when trying to determine the location of an object during the double slit experiment.

The Experiment

Thomas Young designed and conducted an experiment which had shocking results. He was attempting to show that light behaved as waves rather than the accepted notion that light behaved as individual particles. Young knew that waves created interference patterns when they interacted with each other. To test his hypothesis he shined a light source at a barrier with two slits which were fairly close to one another. The light passed through the slits and shone on a screen behind the barrier. If light was acting as a wave he would expect to see an interference pattern on the screen. If light was acting as a particle he would expect to see an image of the the two slits as the particles passed through the barrier.

Image result for double slit experiment — A depiction of Thomas Young’s double slit experiment. Courtesy of curiosity.com

The Results of the experiment

When Young performed his experiment an interference pattern emerged on the screen behind the barrier. This seemed to confirm Young’s idea that light behaved as a wave. In the early part of the 20th century quantum mechanics was born. Scientists like Max Planck, the father of quantum mechanics and Albert Einstein argued that light behaved as both a particle, the photon and a wave. The double slit experiment was done again only this time scientists were able to fire photons one by one at the barrier. In this case one would think that the pattern that emerged would not be an interference pattern but rather a band of light at through each slit. Think of spraying paint though a two slit barrier. Bands would only be present at the site of the slits, similar to a stencil.

Image result for double slit experiment particle results — The top image is what would be expected if light behaved as particles with the photons present directly in line of sight of the slits. The bottom image is what would be expected if light behaved as a wave. The square of the wave function predicts most likely location of the photons. Courtesy of medium.com

This experiment has been repeated many times since Young’s experiment. Scientists have even repeated the experiment using electrons which they could fire one at a time to try to determine wave versus particle behavior. Even when the electrons were fired one by one, eventually an interference pattern emerged. When the electrons are fired one at a time they can not be interacting with other electrons to produce an interference pattern. This means the electrons are interacting with themselves to create an interference pattern.

Here is where it gets weird. In order to for an interference pattern to occur the barrier must have two slits. How can a single electron be aware of another slit? One solution to this question is that the electron splits and goes through each of the two slits and interacts with itself. Another alternative is that the electron does not split but goes through both slits at the same time. Both of these scenarios vary greatly from our everyday experiences dealing with Newtonian mechanics.

With advances in technology we can now perform the experiment using either wave or particle detectors to once and for all determine if an electron behaves as a particle or a wave. Quantum weirdness, however, rears it head once again. If we try to use a detector to see which route, that is which slit, the electron went through, then no interference pattern is detected and the result is consistent with particle behavior. The act of observing, in this case by use of a detector, destroys the interference pattern by collapsing the wave function. The implication is that the act of observing which slits the electrons goes through causes the interference pattern to collapse and the particle pattern to be demonstrated. If, however, no detector is used then the wave interference pattern prevails. These results demonstrate the dual nature of many things once thought of as particles and that observation at the quantum level can have a dramatic impact on the very results one is tying to measure.