Rotational invariance is a property of a physical system or mathematical equation that remains unchanged under rotations. This means that if a system or equation exhibits rotational invariance, it will not be affected by rotations in any way. There are many examples of rotational invariance in physics, but we will focus on two here: the laws of physics and the Schrödinger equation.

The laws of physics are an example of rotational invariance because they do not change when the system is rotated. For example, consider the law of gravitation, which states that the force of gravity between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them. If we rotate the system, the masses of the objects and the distance between them will not change, so the law of gravitation will not be affected by the rotation. This is true for all the laws of physics, including the laws of motion, thermodynamics, and electromagnetism.

Another example of rotational invariance is the Schrödinger equation, which is a fundamental equation in quantum mechanics that describes the time-evolution of a quantum mechanical system. The Schrödinger equation is written as:

iℏ ∂ψ/∂t = Hψ

Where i is the imaginary unit, ℏ is the reduced Planck constant, ∂ψ/∂t is the time derivative of the wave function ψ, and H is the Hamiltonian operator. The Hamiltonian operator represents the energy of the system and is written as:

H = ∑i p^2/(2m) + V(r)

Where p is the momentum operator, m is the mass of the particle, and V(r) is the potential energy.

The Schrödinger equation is rotational invariant because it does not change when the system is rotated. For example, consider a quantum mechanical system that consists of a single particle in a box. If we rotate the box, the wave function of the particle will not change because the potential energy V(r) is independent of the orientation of the box. This means that the Schrödinger equation will remain unchanged under rotations.

In summary, rotational invariance is a property of a system or equation that remains unchanged under rotations. The laws of physics and the Schrödinger equation are two examples of rotational invariance because they do not change when the system is rotated. This property is important in many areas of physics, including quantum mechanics, where it helps to simplify calculations and make predictions about the behavior of systems.