# Degrees of freedom in the energy density formula for black body radiation

• goldbloom55
In summary, the conversation discusses a formula in Quantum Mechanics Demystified for energy density u(\nu,t) and its relation to the theoretical black body radiation experiment. The formula involves the number of degrees of freedom for a given frequency and the average energy per degree of freedom. The concept of degrees of freedom is explained as the number of variables that can be varied in the formula and cannot be reduced through mathematical means. The conversation also clarifies that the degrees of freedom refer to the 3 degrees of position and 3 degrees of momentum in space.

#### goldbloom55

I'm looking for a conceptual explanation of a formula in Quantum Mechanics Demystified introduction. They introduce you to the theoretical black body radiation experiment, where demonstrated how a classical approach leads to the ultraviolet catastrophe.

In the explanation they have the following formula for energy density u($\nu$,t):

u = ( number degrees of freedom for frequency $\nu$) x (average energy per degree of freedom)

My understanding is that that degrees of freedom is the number of variables that you can vary in the formula.

Since the degrees of freedom vary, I'm assuming that under certain temperatures and frequencies, the formula can be simplified and reduced? This would in turn change the degrees of freedom? Please let me know if I'm understanding it correctly.

It is not just a mathematical abstract. They mean the 3 degrees of position in space and the 3 degrees of momentum in space. I may have missed some, but the number of degrees of freedom is not reduceable by math.

## 1. What is the energy density formula for black body radiation?

The energy density formula for black body radiation is given by u = (8πkBT3)/(c3h3), where u is the energy density, kB is the Boltzmann constant, T is the temperature, c is the speed of light, and h is the Planck constant.

## 2. What is the significance of degrees of freedom in the energy density formula for black body radiation?

Degrees of freedom in the energy density formula for black body radiation refer to the number of independent ways in which energy can be distributed among the particles in a system. In the formula, degrees of freedom are represented by the exponent 3 in the term T3. This term takes into account the different ways in which energy can be distributed among the photons in the black body radiation.

## 3. How do degrees of freedom affect the energy density of black body radiation?

Degrees of freedom have a direct impact on the energy density of black body radiation. As the number of degrees of freedom increases, the energy density also increases. This is because with more degrees of freedom, there are more ways in which energy can be distributed among the particles, resulting in a higher energy density.

## 4. Why is the speed of light and Planck constant included in the energy density formula for black body radiation?

The speed of light and Planck constant are included in the energy density formula for black body radiation because they are fundamental constants that describe the behavior of energy and matter at the atomic and subatomic levels. The speed of light is necessary for converting between energy and frequency, while the Planck constant is necessary for converting between energy and wavelength.

## 5. How does the energy density formula for black body radiation relate to the Stefan-Boltzmann law?

The energy density formula for black body radiation and the Stefan-Boltzmann law are closely related. The energy density formula is essentially a simplified version of the Stefan-Boltzmann law, which describes the total amount of radiation emitted by a black body at a given temperature. The energy density formula focuses on the energy density of the radiation, while the Stefan-Boltzmann law takes into account the surface area of the emitting body as well.