Mathematically equation (7-22) is the same differential equation as we had for a particle in a box-the function and its second derivative are proportional. But$\mathbf{\left(}\mathit{\varphi }\mathbf{\right)}$for m₁= 0is a constant and is allowed, whereas such a constant wave function is not allowed for a particle in a box. What physics accounts for this difference?

The quantum number is used to describe the trajectory as well as the movement of the electron in an atom. According to the Pauli Exclusion principal, no electron in an atom can have same set of the quantum numbers.

In case of the hydrogen atom and other particles in the spherical well, no barrier is encountered by varying the azimuthal angle over the complete range of values from 0 to 2. The wave function is not required to be zero that is encountered and the requirement instead is only that wave function at any angle must return to the same value at an angle that is greater. The wave function for m₁ = 0, which has no dependence, already satisfies the condition without forcing anything to be zero everywhere.

Therefore, the wave function m₁ = 0 , which has no dependence, and satisfies the condition without forcing anything to be zero everywhere.