The time dependence of a physical quantity p is given by p = p₀exp(-αt²),where α is a constant and t is the time.The constant α

To find the dimensions of the constant $\alpha$, we use the principle of dimensional homogeneity.

The given equation for the time dependence of the physical quantity $p$ is:
$p=p_0{e}^{-\alpha t^2}$
Here, $t$ represents time, and the term inside the exponential function is $-\alpha t^2$.

In physical equations, the arguments of mathematical functions such as exponentials, logarithms, and trigonometric functions must be dimensionless. This is because these functions can be expanded as infinite power series, and it is only mathematically and physically consistent to add terms of the same dimension (which must be dimensionless).

Therefore, the quantity $\alpha t^2$ must be dimensionless:
$\left[\alpha t^2\right]=1$

This can be written in terms of individual dimensions as:
$\left[\alpha \right]\times \left[t^2\right]=1$

Since $t$ represents time, its dimension is $\left[T\right]$. Substituting this in, we get:
$\left[\alpha \right]\times \left[T^2\right]=1$

Solving for the dimensions of $\alpha$:
$\left[\alpha \right]=\frac{1}{\left[T^2\right]}=\left[T^{-2}\right]$

Hence, the constant $\alpha$ has dimensions $\left[T^{-2}\right]$.