In quantum mechanics, the speed of a wave is not determined by its wavelength. Instead, the speed of a wave is determined by the properties of the medium through which it is propagating. The speed of light, for example, is constant in a vacuum and is denoted by the symbol "c."
In a vacuum, all electromagnetic waves, including light, travel at the speed of light. This means that regardless of their wavelengths (short or long), they all move at the same speed. The speed of light in a vacuum is approximately 299,792,458 meters per second.
However, when light or any other electromagnetic wave travels through a material medium, its speed can be different from the speed of light in a vacuum. This difference is due to the interaction of the wave with the atoms or molecules of the medium.
In such cases, the speed of the wave depends on the refractive index of the medium, which is a measure of how much the speed of light is reduced as it passes through the medium. The refractive index is related to the wavelength of the wave but not in a way that shorter wavelengths always move faster than long ones.
In materials with a higher refractive index, light tends to slow down, regardless of its wavelength. This effect can cause different wavelengths to exhibit different speeds when passing through a medium. This phenomenon is known as dispersion.
In summary, in quantum mechanics, the speed of a wave is not directly determined by its wavelength. The speed of a wave is determined by the properties of the medium through which it is propagating, such as the refractive index. The interaction of the wave with the medium can lead to dispersion, where different wavelengths exhibit different speeds, but this is not a general rule that shorter wavelengths always move faster than long ones.