Rotational Invariance

TL;DR

• A system or equation is rotationally invariant if rotating the system does not change its description or predictions.
• Many fundamental physical laws and equations exhibit this symmetry, so rotating the coordinate frame leaves them unchanged.
• This symmetry is used in physics (including quantum mechanics) to simplify calculations and predictions.

Definition

Rotational invariance is a property of a physical system or mathematical equation that remains unchanged under rotations. If a system or equation exhibits rotational invariance, it is not affected by rotations.

Explanation

When a system is rotationally invariant, rotating the system does not alter the quantities or relations that describe it. For instance, in the law of gravitation the masses of objects and the distance between them remain the same under rotation, so the gravitational law is unaffected. In quantum mechanics, the Schrödinger equation is rotationally invariant when its Hamiltonian does not depend on the orientation, so the time evolution of the wave function is unchanged by rotations.

The Schrödinger equation is written as:

$$i\hbar \frac{\partial \psi}{\partial t} = H\psi$$

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 in the source is given as:

$$H = \sum_i \frac{p^2}{2m} + V(r)$$

where p is the momentum operator, m is the mass of the particle, and V(r) is the potential energy. If V(r) is independent of orientation, the Schrödinger equation remains unchanged under rotations.

Examples

Laws of physics (gravitation)

The law of gravitation states that the force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them. Rotating the system does not change the masses or the distance, so the law is unaffected by rotation. This example applies similarly to other laws such as the laws of motion, thermodynamics, and electromagnetism.

Schrödinger equation (single particle in a box)

The Schrödinger equation

$$i\hbar \frac{\partial \psi}{\partial t} = H\psi$$

is rotationally invariant when the Hamiltonian

$$H = \sum_i \frac{p^2}{2m} + V(r)$$

does not depend on orientation. For a single particle in a box, if the potential energy V(r) is independent of the box’s orientation, the wave function and the Schrödinger equation remain unchanged under rotation.

Use cases

• Rotational invariance is important in many areas of physics, including quantum mechanics, where it helps to simplify calculations and make predictions about the behavior of systems.

Related terms

• Schrödinger equation
• Hamiltonian operator
• Momentum operator
• Potential energy
• Laws of physics
• Gravitation