User:Johnjbarton/sandbox/introduction to eigenstates

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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.^[1] For that reason, the correct description of an eigenstate always includes the particular measurement type. For example, measurement of the $x$ momentum prepares an eigenstate of the $x$ momentum, $p_{x}$ . Repeated measurement of $p_{x}$ momentum for an eigenstate of $p_{x}$ momentum produces consistently repeatable results.

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.^[1]^: 140 For example, measurements of momentum $p_{x}$ 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.^[1]^: 141

For a concrete example, imagine a tall evacuated tube with an electron source (a heated filament with some electron lenses) on the top end. Call the top-bottom direction the $z$ axis. Assign $x$ and $y$ coordinates across the tube. Near the bottom of the tube place a metal plate with a tiny hole. An electron detector, placed along the tube in front of the plate and pointing in to the center, will detect little current. The apparatus has prepared an eigenstate of momentum along the $p_{x}$ and the $p_{y}$ directions. Behind to hole and across the tube, an imaging electron detector will see a spread out circular pattern of electron current called an Airy disk.

Airy-pattern. This numerically calculated image matches the experimental one^[2]^{: fig. 2}

The hole creates an eigenstate of $x$ position and of $y$ position.^[1]^: 142^[3]^: 54 The $p_{x}$ and $p_{y}$ 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.^[2]

Eigenstates and probability

Unlike everyday measurements, individual quantum measurements give random values. No single value has meaning.^{[citation needed]} As more and more identical measurements are performed, some values appear more often that others. Quantum measurements must be performed many times and averaged to produce a definite value. 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 the electron version of the double-slit experiment shown in the video below.

Individual dots appear at seemingly random locations. As more and more dots arrive, the pattern slowly appears. The final pattern is only clear once many events have been averaged.

Recombining Eigenstates

Splitting then recombining eigenstates produces interference patterns.^[5]^: 256 Replace the tiny hole used in the example in the previous section with two tiny holes or slits.

Two slits are illuminated by a plane wave.

The two holes split the incoming eigenstate in two and they recombine on the detector. The combination is not simple addition.

Now the detector shows a dramatic pattern light alternating with dark.^[6]

Eigenstates and eigenvalues

The word "eigenstate" is derives from Schrodinger's use of the word eigenwertproblem ^[7] 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.^[8]^: 126 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.^[1]^: 188 Some experiments, like the Stern-Gerlach experiment shown in the figure, produce results consistent with quantized, discrete eigenvalues.^{[citation needed]}

Eigenstates with multiple indistinguishable particles

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

Saved for refs at the end

However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an electron, given that its position has been determined at a given instant. Likewise, it is impossible to determine the exact location of that particle once its momentum has been measured at a particular instant.^[9] Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. The uncertainty principle also says that eliminating uncertainty about position maximizes uncertainty about momentum, and eliminating uncertainty about momentum maximizes uncertainty about position. A probability distribution assigns probabilities to all possible values of position and momentum. Schrödinger's wave equation gives wavefunction solutions, the squares of which are probabilities of where the electron might be, just as Heisenberg's probability distribution does.^[10]^[11]

Therefore, it became necessary to formulate clearly the difference between the state of something that is uncertain in the way just described, such as an electron in a probability cloud, and the state of something having a definite value. When an object can definitely be "pinned down" in some respect, it is said to be in an eigenstate for that respect. As stated above, when the wavefunction collapses because the position of an electron has been determined, the electron's state becomes an "eigenstate of position", meaning that its position has a known value, an eigenvalue of the eigenstate of position.^[12]

References

^ ^a ^b ^c ^d ^e Messiah, Albert (1966). Quantum Mechanics. North Holland, John Wiley & Sons. ISBN 0486409244.
^ ^a ^b Matteucci, Giorgio; Ferrari, Loris; Migliori, Andrea (2010-09-01). "The Heisenberg uncertainty principle demonstrated with an electron diffraction experiment" (PDF). European Journal of Physics. 31 (5): 1287–1293. doi:10.1088/0143-0807/31/5/027. ISSN 0143-0807.
^ Greiner, Walter (2000). "The Heisenberg Uncertainty Principle". Quantum mechanics. New York: Springer-Verlag. pp. 51–63, 79. ISBN 978-3-540-67458-0. Example 3.5 Position measurement with a slit
^ Bach, Roger; Pope, Damian; Liou, Sy-Hwang; Batelaan, Herman (2013-03-13). "Controlled double-slit electron diffraction". New Journal of Physics. 15 (3). IOP Publishing: 033018. arXiv:1210.6243. Bibcode:2013NJPh...15c3018B. doi:10.1088/1367-2630/15/3/033018. ISSN 1367-2630. S2CID 832961.
^ Born, M.; Wolf, E. (1999). Principles of Optics. Cambridge University Press. ISBN 978-0-521-64222-4.
^ Matteucci, Giorgio; Ferrari, Loris; Migliori, Andrea (2010-09-01). "The Heisenberg uncertainty principle demonstrated with an electron diffraction experiment". European Journal of Physics. 31 (5): 1287–1293. doi:10.1088/0143-0807/31/5/027. ISSN 0143-0807.
^ Schrödinger, Erwin. "Quantisierung als eigenwertproblem." Annalen der physik 385.13 (1926): 437-490.
^ Cite error: The named reference Susskind-Friedman was invoked but never defined (see the help page).
^ Cite error: The named reference Uncertaintyprinciple1 was invoked but never defined (see the help page).
^ Greiner, Walter (2000). Quantum mechanics; (probability distribution). New York: Springer-Verlag. pp. 170, 249, 251, 313. ISBN 978-3-540-67458-0.
^ Gamow, George (1966). Thirty Years That Shook Physics: The Story of Quantum Theory. Mineola, NY: Dover Publications. pp. 3, 90–95, 105, 113–14. ISBN 978-0-486-24895-0.
^ "Energy state". Sci-tech dictionary. 2010.

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)

[messiah-1] Messiah, Albert (1966). Quantum Mechanics. North Holland, John Wiley & Sons. ISBN 0486409244.

[matteucii-2] Matteucci, Giorgio; Ferrari, Loris; Migliori, Andrea (2010-09-01). "The Heisenberg uncertainty principle demonstrated with an electron diffraction experiment" (PDF). European Journal of Physics. 31 (5): 1287–1293. doi:10.1088/0143-0807/31/5/027. ISSN 0143-0807.

[Greiner-3] Greiner, Walter (2000). "The Heisenberg Uncertainty Principle". Quantum mechanics. New York: Springer-Verlag. pp. 51–63, 79. ISBN 978-3-540-67458-0. Example 3.5 Position measurement with a slit

[Bach_Pope_Liou_Batelaan_2013_p=033018-4] Bach, Roger; Pope, Damian; Liou, Sy-Hwang; Batelaan, Herman (2013-03-13). "Controlled double-slit electron diffraction". New Journal of Physics. 15 (3). IOP Publishing: 033018. arXiv:1210.6243. Bibcode:2013NJPh...15c3018B. doi:10.1088/1367-2630/15/3/033018. ISSN 1367-2630. S2CID 832961.

[Born_&_Wolf-5] Born, M.; Wolf, E. (1999). Principles of Optics. Cambridge University Press. ISBN 978-0-521-64222-4.

[6] Matteucci, Giorgio; Ferrari, Loris; Migliori, Andrea (2010-09-01). "The Heisenberg uncertainty principle demonstrated with an electron diffraction experiment". European Journal of Physics. 31 (5): 1287–1293. doi:10.1088/0143-0807/31/5/027. ISSN 0143-0807.

[7] Schrödinger, Erwin. "Quantisierung als eigenwertproblem." Annalen der physik 385.13 (1926): 437-490.

[Susskind-Friedman-8] Cite error: The named reference Susskind-Friedman was invoked but never defined (see the help page).

[Uncertaintyprinciple1-9] Cite error: The named reference Uncertaintyprinciple1 was invoked but never defined (see the help page).

[Uncertaintyprinciple2-10] Greiner, Walter (2000). Quantum mechanics; (probability distribution). New York: Springer-Verlag. pp. 170, 249, 251, 313. ISBN 978-3-540-67458-0.

[Schrödinger1-11] Gamow, George (1966). Thirty Years That Shook Physics: The Story of Quantum Theory. Mineola, NY: Dover Publications. pp. 3, 90–95, 105, 113–14. ISBN 978-0-486-24895-0.

[12] "Energy state". Sci-tech dictionary. 2010.

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