Why are plane waves not possible representations of states in quantum theory?

In summary, the conversation discusses the postulates of quantum mechanics and how they relate to the problem of the free particle. The stationary Schrodinger equation in one dimension is presented and it admits solutions that are not normalizable and thus not in a Hilbert space. However, these states can be used as idealizations and physically realizable scattering states are superpositions of them. The conversation also mentions the Gelfand construction, which justifies the approach of physicists to these matters, and provides resources for further reading on the topic.
  • #1
AxiomOfChoice
533
1
One of the postulates of quantum mechanics, as quoted in any textbook on the subject, is something like the following: "states are vectors in a Hilbert space."

But then they go on to solve the problem of the free particle, which should (I guess) be about the simplest problem one can solve. The associated stationary Schrodinger equation in one dimension looks like

[tex]
-\frac{\hbar^2}{2m} \psi'' = E\psi.
[/tex]

This admits solutions of the form [itex]\psi(x) = Ae^{ikx} + Be^{-ikx}[/itex], where of course [itex]k = \sqrt{2mE}/\hbar[/itex]. But these are NOT normalizable and are therefore not in any sort of Hilbert space, since any vector in a Hilbert space necessarily has finite norm (and can therefore be normalized). So what is going on here?
 
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  • #2
Those states are not physically realizable. Physically realizable scattering states are superpositions of them (wave packets). Nevertheless, they are useful as idealizations.
 
  • #3
jtbell said:
Those states are not physically realizable. Physically realizable scattering states are superpositions of them (wave packets). Nevertheless, they are useful as idealizations.

Okay. So though those states themselves are not in our Hilbert space, certain linear combinations (and I guess we need to talk about [itex]\int[/itex] instead of [itex]\sum[/itex] when we say "linear combinations"...or do we?) of them are?
 
  • #4
Plane waves are not possible representations of states in quantum theory. They are distributions and belong to the dual of the domain of the position and momentum operators and all their powers. This domain is a dense subspace of the Hilbert space, and it's dual is thus much larger than the Hilbert space (for a Hilbert space the topological dual is isomorphic with the Hilbert space itself).

This formulation of quantum theory, called Gelfand construction (or rigged Hilbert space), justifies the quite handwaving approach of physicists to these matters, which goes back to Dirac. A very nice pedagogical introduction can be found in

R. de la Madrid, The role of rigged Hilbert space in quantum mechanics, Eur. J. Phys. 26, 287 (2005) 287
doi:10.1088/0143-0807/26/2/008

A good textbook using this formulation is

L. Ballentine, Quantum Mechanics

and a more formal mathematical representation is given in

Galindo, A., and Pascual, P.: Quantum Mechanics, Springer Verlag, 1990, 2 Vols.
 

1. What are scattering states?

Scattering states refer to the behavior of particles or waves when they encounter obstacles or disturbances in their path. It is a fundamental concept in physics that helps us understand the behavior of matter in various situations.

2. How do scattering states differ from bound states?

Bound states are stationary states of matter where particles are confined to a specific region, while scattering states involve particles that are free to move and interact with their surroundings. Bound states have discrete energy levels, whereas scattering states have a continuous spectrum of energies.

3. What factors affect scattering states?

Several factors can influence scattering states, such as the type of particles or waves involved, the properties of the medium they are traveling through, and the shape and size of the obstacles they encounter. These factors can affect the scattering angle, energy, and intensity of the particles or waves.

4. How are scattering states used in practical applications?

Scattering states have a wide range of applications in different fields, such as materials science, geology, and medical imaging. In materials science, they are used to study the structure and properties of materials. In geology, they help us understand the composition of rocks and minerals. In medical imaging, they are used to create images of internal body structures.

5. How can I visualize scattering states?

There are various ways to visualize scattering states, depending on the type of particles or waves involved. For example, in quantum mechanics, we use mathematical equations and diagrams to represent scattering states. In experiments, we can use detectors to measure the scattering angle and intensity of particles or waves. In simulations, we can use computer programs to create visualizations of scattering processes.

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