User:Johnjbarton/sandbox/measurement in quantum mechanics

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Differences from everyday measurement[edit]

Quantum states[edit]

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. Measurements on the quantum world are sufficiently different that physics uses special words for their results. A quantum system is a physical system viewed at the atomic or subatomic level and governed by quantum mechanics; a quantum state lists the properties and their values for all of the components of a quantum system. The properties include relative locations and velocities, as well as properties charactering internal states. The components of a quantum system are subatomic particles or combinations of them, like atoms and molecules.

Experimental or simulated theoretical measurements determine the values of these properties of a quantum state. Unlike everyday measurements, quantum measurements alter quantum states; the result is called an eigenstate. An eigenstate is a measured quantum state in which at least one quantifiable characteristics such as position or momentum has a determined value.^[1]^: 126

Filtering[edit]

A quantum measurement produces an eigenstate, but more correctly, particular measurements produce particular eigenstates.^[2] 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. The measurement may be a preliminary step performed to create an eigenstate for subsequent work. In that case the measurement may be called preparing an eigenstate.^[2]^: 204

Ordinary objects are measured by comparing them to standard objects, like rulers or protractors or stop watches. Atoms and elementary particles are far too small for direct comparisons, millions of times smaller than a human hair. Ordinary comparisons most often rely on visual comparisons that require light energy to bounce off the object. Light momentum pushes atoms around, changing the very thing we are trying to measure.^[2]^: 141

For these reasons quantum states are filtered to prepare for experiments and the measurements associated with eigenstates uses filtering rather than direct comparison.^[2]^: 196 For example, position measurements may record the intensity of light or current through tiny apertures,^[2]^: 144 sometimes using many thousands of apertures in parallel producing a 2D map of intensity versus position. Momentum measurements might deflect a charged particle beam in a constant magnetic field, collecting only those particles that exit though a slit.^[2]^: 144 Particles with magnetic moments can be filtered with inhomogeneous magnet fields.^[3]

Experimental versus idealized mathematical measurement[edit]

Physicists also use the word "eigenstate" as a synonym for "eigenfunction" or "eigenvector",^[4]^: 506 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.^[2]^: 204

In theoretical calculations a quantum state is "measured" by a mathematical filtering process. A mathematical representation of a measurement device, called an operator, modifies the solutions to, for example, Schrodingers equation producing both an eigenfunction and an associated number called an eigenvalue. These theoretical functions represent pure states and the eigenvalues represent idealized experimental results, as if the raw quantum state were filtered using all possible compatible measurements.^[2]^: 204. To compare to real experiments, the eigenvalues from theories need to be averaged into mixed quantum states, each value weighted by its probability; the result is called the expected value of the measurement according to the theory.^[5]^: 27

References[edit]

^ Susskind, Leonard; Friedman, Art; Susskind, Leonard (2014). Quantum mechanics: the theoretical minimum; [what you need to know to start doing physics]. The theoretical minimum / Leonard Susskind and George Hrabovsky. New York, NY: Basic Books. ISBN 978-0-465-08061-8.
^ ^a ^b ^c ^d ^e ^f ^g ^h Messiah, Albert (1966). Quantum Mechanics. North Holland, John Wiley & Sons. ISBN 0486409244.
^ Castelvecchi, Davide (2022-02-28). "The Stern–Gerlach experiment at 100". Nature Reviews Physics. 4 (3): 140–142. doi:10.1038/s42254-022-00436-4. ISSN 2522-5820.
^ Penrose, Roger (2006). The road to reality: a complete guide to the laws of the universe (8. printing ed.). New York, NY: Knopf. ISBN 978-0-679-45443-4.
^ Schiff, Leonard I. (1995). Quantum mechanics. International series in pure and applied physics (3. ed., 29. print ed.). New York: McGraw-Hill. ISBN 978-0-07-055287-6.

[Susskind-Friedman-1] Susskind, Leonard; Friedman, Art; Susskind, Leonard (2014). Quantum mechanics: the theoretical minimum; [what you need to know to start doing physics]. The theoretical minimum / Leonard Susskind and George Hrabovsky. New York, NY: Basic Books. ISBN 978-0-465-08061-8.

[messiah-2] ^ ^a ^b ^c ^d ^e ^f ^g ^h Messiah, Albert (1966). Quantum Mechanics. North Holland, John Wiley & Sons. ISBN 0486409244.

[3] Castelvecchi, Davide (2022-02-28). "The Stern–Gerlach experiment at 100". Nature Reviews Physics. 4 (3): 140–142. doi:10.1038/s42254-022-00436-4. ISSN 2522-5820.

[4] Penrose, Roger (2006). The road to reality: a complete guide to the laws of the universe (8. printing ed.). New York, NY: Knopf. ISBN 978-0-679-45443-4.

[schiff-5] Schiff, Leonard I. (1995). Quantum mechanics. International series in pure and applied physics (3. ed., 29. print ed.). New York: McGraw-Hill. ISBN 978-0-07-055287-6.

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