Quantum Weirdness The Copenhagen Interpretation Entangled States The

Quantum Weirdness The Copenhagen Interpretation Entangled States The Einstein-Podolsky-Rosen experiment EPR Results and Conclusions Local Realism Bell’s Theorem Applications of Quantum Weirdness Albert Einstein (1879 -1955) The more success the quantum theory has, the sillier it looks. -Albert Einstein Prof. Rick Trebino, Georgia Tech, www. frog. gatech. edu

The Copenhagen Interpretation Quantum mechanics is one of the most successful theories in history. But while its predictions are clear, its interpretation is not. The Copenhagen Interpretation is an interpretation of quantum mechanics. It arose out of discussions between Bohr and Heisenberg in 1927 and was strongly supported by Max Born and Wolfgang Pauli. Max Born (1882 -1970)

The Copenhagen Interpretation 1. A system is completely described by a wave function Y, which represents an observer's knowledge of the system. (Heisenberg) 2. The description of nature is probabilistic. The probability of an event is the mag squared of the wave function related to it. (Max Born) 3. Heisenberg's Uncertainty Principle says it’s impossible to know the values of all of the properties of the system at the same time; properties not known with precision are described by probabilities. 4. Complementarity Principle: matter exhibits a wave-particle duality. An experiment can show the particle-like properties of matter, or wave-like properties, but not both at the same time. (Bohr) 5. Measuring devices are essentially classical devices, and they measure classical properties such as position and momentum. 6. The correspondence principle of Bohr and Heisenberg: the quantum mechanical description of large systems should closely approximate the classical description.

Uncertainty in spin components Recall that the z-component and total angular momentum of the spin are precisely knowable. But the x- and y-components are not. It can be shown that: Using: We find: For electrons, positrons, protons, and neutrons ms = ± 1/2 So, as long as m. S ≠ 0 for a given particle, there’s an Uncertainty relation between the x and y components of its spin. This means that we can measure one component, calling it Sz, (and obtaining ±ħ/2), but doing so randomizes the other two components.

Objections to the Copenhagen Interpretation Many physicists objected to the Copenhagen interpretation’s nondeterministic nature. There were also objections to the vague measurement process that converts probability functions into nonprobabilistic measurements. Some who rejected this interpretation were Albert Einstein, Max Planck, Louis de Broglie, and Erwin Schrödinger. Einstein said to Born: “I, at any rate, am convinced that God does not play dice (with the universe). ”

Superpositions of states Energy Stationary states are stationary. But an atom can be in a superposition of two stationary states, and this state moves. Excited level, E 2 DE = h n Ground level, E 1 where |ai|2 is the probability that the atom is in state i. A superposition means that the atom is vibrating: Vibrations occur at the frequency difference between the two levels.

Wave-Function Collapse It’s our lack of knowledge of which state a system is in that puts it into a superposition state: Making a measurement of the energy of the above state will produce either E 1 or E 2, with probabilities |a 1|2 and |a 2|2, respectively. In the Copenhagen interpretation, the state collapses to the measured one, and the unobserved state is removed from further consideration. This is called wave-function collapse. Unmeasured possible states simply disappear from sight like losing lottery tickets.

Schrödinger’s Cat To reveal what he considered its absurdity, Schrodinger proposed (but fortunately never implemented!) putting a cat in a sound-proof box and killing it with a ½ probability. Before we open the box, is the cat alive or dead? Even though the cat may feel otherwise, quantum mechanics says the cat is both! It’s in a superposition of “alive” and “dead. ” Making a measurement on the system (peaking into the box) collapses the cat’s state to either “alive” or “dead. ”

The EPR Paradox It seems that our consciousness plays a role in quantum mechanics. Einstein became uneasy about such implications and, in later years, organized a rearguard action against quantum mechanics. His question, “Do you really think the moon isn't there if you aren't looking at it? ” highlights the depths of his distaste for the role of the consciousness. His strongest counter-argument was a paradoxical implication of quantum mechanics now known as the Einstein-Podolsky-Rosen (EPR) Paradox.

The Einstein-Podolsky-Rosen Paper Einstein believed that, while quantum mechanics could be used to make highly accurate statistical predictions about experiments, it’s an incomplete theory of physical reality. In 1935, Einstein, working with physicists Boris Podolsky and Nathan Rosen, published the paper, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? ” In this paper, they devised a clever thought experiment that “beat” the Uncertainty Principle. So they concluded that there must be more going on than quantum mechanics knew about, concluding: The quantum-mechanical description of reality given by the wave function is not complete, that is, there must be Hidden Variables that we don’t know about and hence don’t measure that cause the uncertainty.

Hidden Variables Suppose that you’re modeling a baseball pitch, but you don’t know about air. Air, combined with the ball’s spin, causes it to curve, and variations in air pressure cause it to wobble away from your theoretically perfect parabolic path. You’d find that the pitch arrives in a somewhat random position. Your theory is incomplete, and the air pressure vs. position is a hidden variable. Locations of actual pitches with the same initial position and velocity Theoretically calculated location for the initial position and velocity Strike zone

Imagine a pair of particles whose quantum spins are known to be opposite. We can actually know that the total spin S of the twoparticle system is zero if it’s in an S = 0 or “singlet” state. So one is spin-up, and the other is spindown, but we don’t know which is which. EPR: Entangled States Initial twoparticle system with zero spin Two particles emerging from initial system with opposite spins Now separate them and measure the spin of one particle. Because they were paired, they have a combined entangled wave function:

Entangled States Initial twoparticle system But we’re free to choose which component of the spin we’d like to measure. Let’s now pick a perpendicular direction. We can write the same statement about that direction also: Two particles emerging from initial system Of course, Quantum Mechanics says we cannot make precise measurements of both components; making one measurement perturbs the other. In any case, making a measurement of either component of one particle’s spin determines the other. When the measurement is made, the wave function collapses: or

The EPR Paradox Now do something really interesting: Measure the vertical spin component of particle A and the horizontal spin component of particle B. Because the particle A measurement determines both particles’ vertical spin components, and the particle B measurement determines both particles’ horizontal spin components, haven’t we determined two components of each particle’s spin? And beaten the Quantum Mechanics? Initial two-particle system Two particles emerging from initial system

EPR Paradox Initial twoparticle system If this works, then Quantum Mechanics is incomplete, that is, it’s actually possible to make precise measurements if we’re clever, and there’s more going on than is in Quantum Mechanics. Two particles emerging from initial system This would be an argument for the existence of Hidden Variables— additional quantities that exist and affect systems, but we just don’t know about yet and so can’t control them. Einstein, perhaps thinking that he’d nailed Quantum Mechanics

Alas, Einstein’s trick doesn’t work! Measuring the vertical-spin component of particle A collapses both particles’ vertical-spin-component states, as predicted. But, in the process, it randomizes both particles’ horizontal-spin components! Measuring A’s vertical spin is just like measuring B’s also! Even though we never touched particle B! Quantum Mechanics wins! Quantum Mechanics 1. Einstein 0. But now you might wonder: Information can’t travel faster than the speed of light. Suppose we let the particles travel many meters (i. e. , many nanoseconds for light) apart, and we make the measurements only picoseconds apart in time, so there isn’t time for the information from the measurement on particle A to reach particle B in time to mess up its measurement. That should save Einstein’s idea. But it doesn’t! This information appears to travel infinitely fast. So this appears to invalidate Einstein’s beloved Special Relativity! Quantum Mechanics wins again! Quantum Mechanics 2. Einstein 0.

Implicit assumptions of EPR The principle of reality: individual particles possess definite properties even when they’re not being observed. The locality principle: information from a measurement in one of two isolated systems cannot produce real change in the other, especially superluminally (faster than c). Taken together, these two seemingly obvious principles imply an upper limit to the degree of co-ordination possible between isolated systems or particles. Interestingly, they both turn out to be wrong.

Local realism is out. John Bell showed in a 1964 paper entitled "On the Einstein Podolsky Rosen paradox, ” that local realism leads to a series of requirements— known as Bell’s inequalities. John Bell (1928 -1990) Alain Aspect has performed numerous beautiful experiments, proving conclusively that our universe violates Bell’s Inequalities big time. And quantum mechanics explains the effects quite nicely. Alain Aspect (1947 -)

Serious Quantum Weirdness EPR assumed that the particles had spin in the first place (Reality). And that such information couldn’t travel infinitely fast (Locality). It appears that particles simply do not have properties until we measure them. It isn’t merely a matter of ignorance. And such information can travel superluminally. These effects are now known as nonlocal behavior, quantum weirdness (and colloquially as spooky action at a distance).

The EPR paradox (which isn’t in the end a paradox) has deepened our understanding of quantum mechanics by exposing the fundamentally nonclassical and unintuitive characteristics of the measurement process. Post-EPR Analysis EPR experiments show that a "measurement" can be performed on a particle without disturbing it directly by performing a measurement on a distant entangled particle.

The Copenhagen Interpretation lives! According to the Copenhagen interpretation, physics depends only on the outcomes of measurements. One-slit pattern Two-slit pattern We can determine where the photon hits the screen by noting a flash. The Copenhagen interpretation rejects arguments about where the photon was between the times it was emitted in the apparatus and when it flashed on the screen.

Alternatives to the Copenhagen Interpretation and its weirdness Many Worlds Interpretation The Many Worlds Interpretation says the wave. Every possible outcome of every function is real, but it denies measurement exists in its own "world“. the reality of wave-function collapse. This implies that all So there’s a very large—perhaps possible alternative infinite—number of universes, and histories and futures are real everything that could possibly have —each representing an happened in our past, but didn't, has actual "world" (or "universe"). in fact occurred in the past of some other universe or universes.

Alternatives to the Copenhagen Interpretation and its weirdness The Guide Wave Interpretation In 1927, Louis de Broglie suggested that the Schrodinger wavefunction was a real function that guided real particles along their paths. In 1952, David Bohm envisioned that the wave-function included a form of energy not known to classical physics, what he called the "quantum potential" or "pilot wave. " In the two-slit experiment, the pilot wave would exist through both slits and guide real particles through the slits to obtain an interference pattern. Although the de Broglie-Bohm interpretation does state that there is a real particle following a real path, the statistical nature of the wavefunction and the Heisenberg uncertainty principle remain in effect, and only probabilities for the location of particles can be determined. But to account for quantum weirdness, disturbance of the pilot wave had to propagate instantaneously.

Applications of Quantum Weirdness Technologies relying on quantum entanglement are now being developed. In quantum cryptography, entangled particles are used to transmit signals that cannot be eavesdropped upon without leaving a trace. In quantum computation, entangled quantum states are used to perform computations in parallel, which may allow certain calculations to be performed much more quickly than they ever could be with classical computers.

This presentation was a project by former GT Modern Physics student, Weston Aenchbacher. I’ve modified it significantly for our class. Copenhagen, Denmark