How to handle ontinuos spectrum

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In summary, the wave function in Gasiorowicz's textbook, expressed as psi(x) = Sigma[C_n X U_n(x)] + Integral[dE C(E) X u_E(x)], represents both the discrete spectrum and the continuous spectrum. The discrete spectrum is represented by the first term, while the second term represents the continuous spectrum. Examples of continuous spectrum can be found in textbooks such as Griffiths, particularly in systems with both bound and unbound states, such as the hydrogen atom. The unbound states correspond to free particles and form a continuous spectrum.
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Homework Statement

In Gasiorowicz's textbook, he provided the schemetic expession of the wave function(95page) ; psi(x) = Sigma[C_n X U_n(x)] + Integral[dE C(E) X u_E(x)]
("X" means multiplication symbol)
In this equation, I think that 1st term(sigma) is the case of the discrete spectrum and 2nd term(integral) is the case of the continuous spectrum.
Textbooks explain many example of bound state(i.e. state having discrete spectrum), but most of them didn't explain about continuos spectrum. I am curious about it.
Would you tell me about refences containing this problem, or tell me some examples of the continuos spectrum?

Homework Equations

psi(x) = Sigma[C_n X U_n(x)] + Integral[dE C(E) X u_E(x)]

The Attempt at a Solution

I tried to find examples of continuos spectrum in many textbooks, but I failed.
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  • #2
It's indeed the most general expression you can write down.

Take for instance the case of a free particle. In that case the energy eigenfunctions are labeled by the wave number, k, which takes on continuous values. You should be able to find this example in pretty much any introductory book... (Griffiths for example)

In systems where there are both bound and unbound states, it are these unbound states which correspond to free particles - and hence the continuous spectrum.

Take for example the hydrogen atom. The lowest energy state has a (relative) energy of -13.6 eV. There are infinite number of bound states with an energy <0 eV. But there are also unbound states, namely those with an energy larger than 0 eV and these states form a continuous spectrum. So formally, these should be taken into account since all eigenstates together form a complete basis of the Hilbert space.

1. What is a continuous spectrum?

A continuous spectrum refers to the uninterrupted range of wavelengths of electromagnetic radiation emitted by a source, such as a blackbody. It contains all colors of the visible spectrum and extends into the ultraviolet and infrared regions.

2. How do you handle a continuous spectrum?

To handle a continuous spectrum, you need to first determine the range of wavelengths present in the spectrum. This can be done by using a spectrometer or a prism to separate the different wavelengths of light. Once the range is identified, you can analyze the spectrum to study the properties of the source.

3. What types of sources produce continuous spectra?

Continuous spectra are produced by thermal sources such as stars, planets, and heated objects. They can also be produced by non-thermal sources, such as fluorescent lights, which emit a continuous spectrum with distinct emission lines superimposed.

4. How is a continuous spectrum different from a line spectrum?

A continuous spectrum is a broad band of colors with no gaps or breaks, while a line spectrum is composed of discrete lines of specific frequencies. A continuous spectrum is produced by a source that emits radiation at all wavelengths, while a line spectrum is produced by a source that emits at specific wavelengths only.

5. What can we learn from studying a continuous spectrum?

By studying a continuous spectrum, we can learn about the temperature and composition of the emitting source. The shape of the spectrum can also reveal information about the physical conditions and processes occurring in the source. Additionally, the presence or absence of certain wavelengths can help identify the chemical elements present in the source.

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