User:Johnjbarton/sandbox/introduction to eigenstates

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This article is a non-technical introduction to the subject. For the main encyclopedia article, see Eigenstates.

The physics word eigenstate expresses ideas about the nature of the atomic and subatomic world. Our understanding of this world, a world governed by quantum mechanics, differs from our daily experience in many interesting ways.

Eigenstates cannot be seen

In the everyday world, it is natural and intuitive to think of every object having simultaneously a definite position, a definite momentum; we expect that measurements of these objects will give a definite value and we can assign that measurement a definite time. We expect careful measurements of objects to leave them unaltered; we expect each object to be unique at least in some tiny way if examined carefully enough. Quantum mechanical objects do not fulfill these expectations.

In the quantum world we cannot directly sense the objects. Our only information comes from measurements. The results of measurements on the quantum world have unexpected and interesting properties. Physics uses the unusual word eigenstate to express the unusual character of these quantum measurements.

Eigenstates from measurements

A quantum measurement produces an eigenstate, but more correctly, particular measurements produce particular eigenstates. For that reason, the correct description of an eigenstate always includes the particular measurement type. For example, measurement of $p_{x}$, the $x$ momentum, prepares an eigenstate of the $p_{x}$. Repeated measurement of momentum $p_{x}$ for an eigenstate of $p_{x}$ momentum produces consistently repeatable results.

Physicists also use the word "eigenstate" as a synonym for "eigenfunction" or "eigenvector", mathematical entities used to describe experimental observations. These theoretical entities represent idealized measurements. The results of theoretical calculations rarely directly relate to observations; theoretical results typically need to be combined and averaged before comparisons.

Eigenstates and the uncertainty principle

Unlike measurements of everyday objects, certain pairs of quantum measurements interact, with one such measurement altering eigenstates produced by the other member of the pair. For example, measurements of $p_{x}$, the momentum along the $x$ axis, creates an eigenstate of the $p_{x}$ momentum. Subsequent measurements of position along the $x$ axis, create an eigenstate of $x$ position. Going back and measuring $p_{x}$ momentum again, on this eigenstate of $x$ position, will not agree with the value before the $x$ position measurement. This is known as the uncertainty principle.

For a concrete example, the experiment of Matteucci, Ferrari, and Migliori, sent a parallel electron beam towards a tiny hole in a metal film mounted in a vacuum chamber. The parallel beam has very little momentum in directions across the beam: the apparatus has prepared an eigenstate of momentum in the two perpendicular directions. The hole creates an eigenstate of position for the two directions across the beam. Behind the metal film an electron microscope enlarges the image of the electron current emerging from the hole. Instead of a simple bright spot, the experiment shows a spread out circular pattern of electron current called an Airy disk.

Airy-pattern. This numerically calculated image matches the experimental result of Matteucci, Ferrari, and Migliori,

The incoming momentum eigenstate has been altered: the position measurement with the hole created motion perpendicular to the tube. The diameter of the circular pattern matches predictions of the uncertainty principle.

Recombining Eigenstates

Splitting then recombining eigenstates produces interference patterns. Replace the tiny hole used in the example in the previous section with two tiny holes or slits. The two holes split the incoming eigenstate in two and they recombine on the detector. The combination is not the sum of intensity of two holes. Instead the detector shows a dramatic pattern light alternating with dark.

Schematic diagram of the Bach et al. experiment. Two slits are illuminated by a parallel electron beam.

Experimental electron double slit diffraction pattern. Across the middle of the image the intensity alternates from high to low showing interference in the signal from the two slits. From Roger Bach et al 2013 New J. Phys. 15 033018 Figure 3 cropped to top frame.

As a concrete example, the experiment of Bach et al. sent a parallel electron beam into a metal film with two tiny holes. As illustrated in the schematic diagram the holes are close enough to allow the interaction of the beam altered by the holes.

Eigenstates and probability

Unlike everyday measurements, individual quantum measurements give random values. Large numbers of individual measurements must be combined. As more and more identical measurements are performed, some values appear more often that others. The state part of eigenstate signals this difference: in the quantum world measurements work on a quantum state rather than on definite objects.

For example, consider this electron version of the double-slit experiment discussed in the previous section. The alternating pattern of high and low intensity results from individual events appearing as single dots, seemingly randomly arranged, until many thousand of dots add up to pattern.

Electron double slit diffraction pattern built up dot by dot.

A video of the same experiment shows the build up of the pattern:

Matter wave double slit diffraction pattern building up electron by electron. Each white dot represents a single electron hitting a detector; with a statistically large number of electrons interference fringes appear.

The final pattern is only clear once many events have been averaged.

Eigenstates and eigenvalues

The word "eigenstate" is derives from Schrodinger's use of the word eigenwertproblem in the title of his first paper on the subject; his own translation of this word in his collected works was "problem of proper values". An eigenstate is the measured state of some object possessing quantifiable characteristics such as position, momentum, etc. The state being measured and described must be observable (i.e. something such as position or momentum that can be experimentally measured either directly or indirectly). Theoretical models of quantum phenomena describe eigenstates in terms of eigenfunctions with an an associated value called an eigenvalue. While many eigenfunctions combine to create an eigenstate, each experimental observation corresponds to a single eigenvalue. Some experiments, like the Stern-Gerlach experiment shown in the figure, produce results consistent with quantized, discrete eigenvalues.

Stern–Gerlach experiment: Silver atoms traveling through an inhomogeneous magnetic field, and being deflected up or down depending on their spin; (1) furnace, (2) beam of silver atoms, (3) inhomogeneous magnetic field, (4) classically expected result, (5) observed result

Eigenstates with multiple indistinguishable particles

Multiple particle bound systems like atoms and molecules have both internal eigenstates, call excited states. and eigenstates associated with their center of mass motion, called matter waves. Quantum particles are indistinguishable, altering the statistical properties of collections of particles.

See also

• Superposition principle
• List of textbooks on classical and quantum mechanics

References

Related reading:

• For wave mechanics
  • Messiah, Albert (1961). "Chapter 5". Quantum mechanics. New York: Dover Publications. p. 162. ISBN 978-0-486-40924-5. {{cite book}}: ISBN / Date incompatibility (help)