# Waves, Energy, Blackbodies and Modes

• Quelsita
In summary, a mode is a particular configuration of the electromagnetic field that can exist within a cavity, and will vary with frequency. Thermal equilibrium requires consideration of the mode density, which depends on the energy of the radiation field.
Quelsita
I am going over some notes and am trying to fit some pieces together. For some reason I keep confusing myself as to what exactly a "mode" is. Is a mode a wave? or a frequency?

Also, how does a mode relate to the degrees of freedom for a particle in a system?

Thanks!

In the context of blackbody (cavity) radiation, a mode is an eigenstate of the radiation field within the cavity. That is, it's a configuration of the electromagnetic field that can exist stably within the cavity: for simple cavitygeometries, analytic expressions for the modes can be written down fairly easily (sines and cosines, bessel functions, hermite polynomials ,etc). Modes will vary with frequency, and can also have different spatial profiles.

Each mode can also be considered a degree of freedom of the electromagnetic field- an arbitrary field can be decomposed into a mode distribution.

Does that help?

Andy Resnick said:
a mode is an eigenstate of the radiation field within the cavity. That is, it's a configuration of the electromagnetic field that can exist stably within the cavity.
That makes sense but "eigenstate" threw me off a bit. So, if an eigenvalue is a scalar, then that just means an eigenstate for, say a photon, would be every possible path it can take?
This is why a mode is a degree of freedom, because it describes the possible configurations?

Andy Resnick said:
Modes will vary with frequency, and can also have different spatial profiles.
So, if you consider a blackbody radiating energy like a standing wave, it has a certain amount of modes possible which increase with frequency.
I know that modes have equal energy and as frequency increases, the energy becomes infinitely large but how does a degree of freedom have energy?

Am I making sense, or just confusing things further?

Forget about the photon picture for now- we are discussing the field *within* a cavity

http://en.wikipedia.org/wiki/Transverse_mode (for example pictures.)
http://en.wikipedia.org/wiki/Longitudinal_mode (for wavelength/frequency info)

Each of the transverse modes are the same wavelength and frequency. If you like, they are standing wave patterns- just like the longitudinal modes are standing wave patterns. For a cubic cavity, the mode structure is very simple- sines and cosines in all three dimensions, with an additional factor of 2 for polarizations.

The missing piece (so far) from this discussion is thermal equilibrium- what is the mode density for a radiation field at thermal equilibrium? As you point out, with increasing frequency there is increasing energy, so we have to take into account the energy dependence of the modes.

http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/rayj.html#c2
http://hyperphysics.phy-astr.gsu.edu/Hbase/mod6.html

## 1. What are waves and how do they transfer energy?

Waves are a form of energy that travels through a medium or space. They transfer energy by causing particles in the medium to vibrate, which then transfers energy to nearby particles.

## 2. What is the difference between mechanical and electromagnetic waves?

Mechanical waves require a medium to travel through, such as water or air, while electromagnetic waves can travel through a vacuum. Mechanical waves also transfer energy by causing particles to vibrate, while electromagnetic waves transfer energy through the oscillation of electric and magnetic fields.

## 3. What is a blackbody and why is it important in understanding energy?

A blackbody is an object that absorbs all radiation that falls on it and emits thermal radiation based on its temperature. It is important in understanding energy because it is a theoretical concept used to explain the behavior of real objects and their interactions with electromagnetic radiation.

## 4. How do modes relate to energy in a system?

Modes refer to the different ways in which energy can be stored in a system, such as in the form of kinetic energy, potential energy, or thermal energy. By understanding the different modes of energy, we can better analyze and describe the behavior of a system.

## 5. How do scientists measure the energy of waves?

Scientists measure the energy of waves using various instruments such as spectrometers, calorimeters, and detectors. These instruments measure the intensity, frequency, and wavelength of the wave to determine its energy. They can also use mathematical equations and models to calculate the energy of waves.

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