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## Summary:

- In some equations, it seems that energy is proportional to mass, and in others, it seems like energy is inversely proportional to mass. Why?

## Main Question or Discussion Point

Hello. I'm a graduate student in electrical engineering, and I'm taking a class in semiconductor physics. My professor has used this equation for the energy of an electron in a semiconductor:

$$E=\frac{\hbar^2 k^2}{2m^*}$$

This seems to imply that energy is inversely proportional to mass, i.e., a heavier particle has less energy. However, we all know the following equation for kinetic energy from high school physics:

$$E=\frac{1}{2}mv^2$$

That equation actually indicates the opposite: energy is proportional to mass, and a heavier particle has more energy. Moreover, it seems like these two equations are derived from each other, using ##p=mv## and ##p=\hbar k##.

$$E=\frac{1}{2}mv^2=\frac{mv^2}{2}=\frac{m^2v^2}{2m}=\frac{p^2}{2m}=\frac{\hbar^2 k^2}{2m}$$

This is somewhat confusing, that these two equations are derived from each other, and yet they seem to be saying opposite things. So which one is true? Do heavier particles have more energy, or less energy? Or are both statements true, in different situations?

$$E=\frac{\hbar^2 k^2}{2m^*}$$

This seems to imply that energy is inversely proportional to mass, i.e., a heavier particle has less energy. However, we all know the following equation for kinetic energy from high school physics:

$$E=\frac{1}{2}mv^2$$

That equation actually indicates the opposite: energy is proportional to mass, and a heavier particle has more energy. Moreover, it seems like these two equations are derived from each other, using ##p=mv## and ##p=\hbar k##.

$$E=\frac{1}{2}mv^2=\frac{mv^2}{2}=\frac{m^2v^2}{2m}=\frac{p^2}{2m}=\frac{\hbar^2 k^2}{2m}$$

This is somewhat confusing, that these two equations are derived from each other, and yet they seem to be saying opposite things. So which one is true? Do heavier particles have more energy, or less energy? Or are both statements true, in different situations?