Final answer:
The equation λ=h/m²v^-1 is dimensionally incorrect because its dimensions do not match the dimensions of the legitimate wavelength equation λ = h/mv, where each of the variables has specific and consistent physical dimensions.
Step-by-step explanation:
To check whether the equation λ=h/m²v^-1 is dimensionally correct, we need to verify that both sides of the equation have the same dimensions.
The proper equation for the wavelength of a particle, according to Planck's equation, is λ = h/mv where λ is the wavelength, h is Planck's constant, m is the particle's mass, and v is the particle's velocity.
The dimensions of Planck's constant are ML²T^-1. The mass m has dimensions of M, and the velocity v has dimensions of LT^-1. When we divide h by m multiplied by v, we get ML²T^-1 / (M*LT^-1), simplifying to L, which is the dimension of wavelength.
Comparing this with the given equation, the dimension of m² would be M², which does not match the correct formula's dimension for mass. Similarly, the inverse of velocity, v^-1, would have dimensions of T/L, which is not present in the correct formula.
Therefore, the equation λ=h/m²v^-1 is dimensionally incorrect because the dimensions on the left side (wavelength) do not match the dimensions on the right side as per the correct formula.