Superposition and Schrodinger’s Equation+Cat

Quantum Mechanics:

  • quantum mechanics is to the microscopic world what classic mechanics and calculus is to the macroscopic world
  • it is the operational process of calculating quantum physics phenomenon
  • its primary task is to bring order and prediction to the uncertainty of the quantum world, its main tool is Schrodinger’s equation
The field of quantum mechanics concerns the description of phenomenon on small scales where classical physics breaks down. The biggest difference between the classical and microscopic realm, is that the quantum world can be not be perceived directly, but rather through the use of instruments. And a key assumption to an quantum physics is that quantum mechanical principles must reduce to Newtonian principles at the macroscopic level (there is a continuity between quantum and Newtonian mechanics).Quantum mechanics was capable of bringing order to the uncertainty of the microscopic world by treatment of the wave function with new mathematics. Key to this idea was the fact that relative probabilities of different possible states are still determined by laws. Thus, there is a difference between the role of chance in quantum mechanics and the unrestricted chaos of a lawless Universe.

Every quantum particle is characterized by a wave function. In 1925 Erwin Schrodinger developed the differential equation which describes the evolution of those wave functions. By using Schrodinger equation, scientists can find the wave function which solves a particular problem in quantum mechanics. Unfortunately, it is usually impossible to find an exact solution to the equation, so certain assumptions are used in order to obtain an approximate answer for the particular problem.

  • the key difference between quantum and classical mechanics is the role of probability and chance
  • quantum objects are described by probability fields, however, this does not mean they are indeterminit, only uncertain
The difference between quantum mechanics and newtonian mechanics is the role of probability and statistics. While the uncertainty principle means that quantum objects have to be described by probability fields, this doesn’t mean that the microscopic world fails to conform to deterministic laws. In fact it does. And measurement is an act by which the measurer and the measured interact to produce a result. Although this is not simply the determination of a preexisting property.The quantum description of reality is objective (weak form) in the sense that everyone armed with a quantum physics education can do the same experiments and come to the same conclusions. Strong objectivity, as in classical physics, requires that the picture of the world yielded by the sum total of all experimental results to be not just a picture or model, but identical with the objective world, something that exists outside of us and prior to any measurement we might have of it. Quantum physics does not have this characteristic due to its built-in indeterminacy.

For centuries, scientists have gotten used to the idea that something like strong objectivity is the foundation of knowledge. So much so that we have come to believe that it is an essential part of the scientific method and that without this most solid kind of objectivity science would be pointless and arbitrary. However, the Copenhagen interpretation of quantum physics (see below) denies that there is any such thing as a true and unambiguous reality at the bottom of everything. Reality is what you measure it to be, and no more. No matter how uncomfortable science is with this viewpoint, quantum physics is extremely accurate and is the foundation of modern physics (perhaps then an objective view of reality is not essential to the conduct of physics). And concepts, such as cause and effect, survive only as a consequence of the collective behavior of large quantum systems.


Schrodinger’s Cat and Quantum Reality:

  • an example of the weirdness of the quantum world is given by the famous Schrodinger cat paradox
In 1935 Schrodinger, who was responsible for formulating much of the wave mechanics in quantum physics, published an essay describing the conceptual problems in quantum mechanics. A brief paragraph in this essay described the, now famous, cat paradox.
  • the paradox is phrased such that a quantum event determines if a cat is killed or not
  • from a quantum perspective, the whole system state is tied to the wave function of the quantum event, i.e. the cat is both dead and alive at the same time
One can even set up quite ridiculous cases where quantum physics rebells against common sense. For example, consider a cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat). In the device is a Geiger counter with a tiny bit of radioactive substance, so small that perhaps in the course of one hour only one of the atoms decays, but also, with equal probability, perhaps none. If the decay happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The wave function for the entire system would express this by having in it the living and the dead cat mixed or smeared out in equal parts.

  • the paradox in some sense is not a paradox, but instead points out the tension between the microscopic and macroscopic worlds and the importance of the observer in a quantum scenario
  • quantum objects exist in superposition, many states, as shown by interference
  • the observer collapses the wave function
It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a “blurred model” for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks. We know that superposition of possible outcomes must exist simultaneously at a microscopic level because we can observe interference effects from these. We know (at least most of us know) that the cat in the box is dead, alive or dying and not in a smeared out state between the alternatives. When and how does the model of many microscopic possibilities resolve itself into a particular macroscopic state? When and how does the fog bank of microscopic possibilities transform itself to the blurred picture we have of a definite macroscopic state. That is the collapse of the wave function problem and Schrodinger’s cat is a simple and elegant explanation of that problem.

Macroscopic/Microscopic World Interface:

  • events in the microscopic world can happen *without* cause = indeterminacy
  • phenomenon such as tunneling shows that quantum physics leaks into the macroscopic world
The macroscopic world is Newtonian and deterministic for local events (note however that even the macroscopic world suffers from chaos). On the other hand, the microscopic quantum world radical indeterminacy limits any certainty surrounding the unfolding of physical events. Many things in the Newtonian world are unpredictable since we can never obtain all the factors effecting a physical system. But, quantum theory is much more unsettling in that events often happen without cause (e.g. radioactive decay).Note that the indeterminacy of the microscopic world has little effect on macroscopic objects. This is due to the fact that wave function for large objects is extremely small compared to the size of the macroscopic world. Your personal wave function is much smaller than any currently measurable sizes. And the indeterminacy of the quantum world is not complete because it is possible to assign probabilities to the wave function.

But, as Schrodinger’s Cat paradox show us, the probability rules of the microscopic world can leak into the macroscopic world. The paradox of Schrodinger’s cat has provoked a great deal of debate among theoretical physicists and philosophers. Although some thinkers have argued that the cat actually does exist in two superposed states, most contend that superposition only occurs when a quantum system is isolated from the rest of its environment. Various explanations have been advanced to account for this paradox–including the idea that the cat, or simply the animal’s physical environment (such as the photons in the box), can act as an observer.

The question is, at what point, or scale, do the probabilistic rules of the quantum realm give way to the deterministic laws that govern the macroscopic world? This question has been brought into vivid relief by the recent work where an NIST group confined a charged beryllium atom in a tiny electromagnetic cage and then cooled it with a laser to its lowest energy state. In this state the position of the atom and its “spin” (a quantum property that is only metaphorically analogous to spin in the ordinary sense) could be ascertained to within a very high degree of accuracy, limited by Heisenberg’s uncertainty principle.

  • decoherence prevents a macroscopic Schrodinger cat paradox
  • new technology allows the manipulation of objects at the quantum level
  • future research will investigate areas such as quantum teleportation and quantum computing
The workers then stimulated the atom with a laser just enough to change its wave function; according to the new wave function of the atom, it now had a 50 percent probability of being in a “spin-up” state in its initial position and an equal probability of being in a “spin-down” state in a position as much as 80 nanometers away, a vast distance indeed for the atomic realm. In effect, the atom was in two different places, as well as two different spin states, at the same time–an atomic analog of a cat both living and dead.The clinching evidence that the NIST researchers had achieved their goal came from their observation of an interference pattern; that phenomenon is a telltale sign that a single beryllium atom produced two distinct wave functions that interfered with each other.

The modern view of quantum mechanics states that Schrodinger’s cat, or any macroscopic object, does not exist as superpositions of existence due to decoherence. A pristine wave function is coherent, i.e. undisturbed by observation. But Schrodinger’s cat is not a pristine wave function, its is constantly interacting with other objects, such as air molecules in the box, or the box itself. Thus a macroscopic object becomes decoherent by many atomic interactions with its surrounding environment.

Decoherence explains why we do not routinely see quantum superpositions in the world around us. It is not because quantum mechanics intrinsically stops working for objects larger than some magic size. Instead, macroscopic objects such as cats and cards are almost impossible to keep isolated to the extent needed to prevent decoherence. Microscopic objects, in contrast, are more easily isolated from their surroundings so that they retain their quantum secrets and quantum behavior.

 

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One thought on “Superposition and Schrodinger’s Equation+Cat

  1. Pingback: Creating Options For Computer Optimised Schrödinger Solutions | Physics News Blog

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