Understanding the Difference between del.A and A.del

The Basics of Dot Product

The dot product or scalar product is an operation that takes two equal-length vectors and returns a single number. For any two vectors A and B, the dot product A⋅B is calculated by multiplying the corresponding components of the vectors and summing them. A key property of the dot product is that it is commutative, meaning A⋅B = B⋅A.

What is the del Operator?

The del operator, represented by the symbol ∇, is a vector differential operator used in physics. ∇ represents the gradient of a function and, rather than being a single operator, is the vector of the partial derivatives of the function with respect to each independent variable. In cartesian coordinates, ∇ can be represented as: ∇ = î∂/∂x + ĵ∂/∂y + k̂∂/∂z Where î, ĵ, k̂ are the unit vectors along the x, y and z axes respectively.

Divergence of a Vector Field

When the del operator acts on a vector field A, written as ∇⋅A, it represents the divergence of that vector field. The divergence measures how a vector field tends to diverge or converge at a given point - it quantifies the net outflow of a vector field through a surface at a point. In cartesian coordinates, the divergence of A is given by: ∇⋅A = ∂Ax/∂x + ∂Ay/∂y + ∂Az/∂z Where Ax, Ay, Az are the components of the vector field A along each axis.

The del Dot Operator Acting on a Vector

When a vector acts on the del operator, written as A⋅∇, it results in a differential operator rather than representing a physical quantity like divergence. In cartesian coordinates, A⋅∇ is expressed as: A⋅∇ = Ax∂/∂x + Ay∂/∂y + Az∂/∂z Where A is now treated as a vector of constant coefficients rather than a vector field. This operator gives the directional derivative of a function in the direction of A.

Commutativity and Physical Meaning

While the dot product is commutative in the abstract algebraic sense, ∇⋅A and A⋅∇ are not interchangeable and do not represent the same physical quantity. ∇⋅A has a clear physical meaning as the divergence, representing how a vector field spreads out from a point. On the other hand, A⋅∇ is simply a differential operator with no clear physical significance on its own in classical physics contexts.

Applications in Classical and Quantum Mechanics

In classical physics fields like electromagnetism, only ∇⋅A is commonly used to represent physically meaningful quantities like divergence. A⋅∇ on its own holds no useful meaning. However, in quantum mechanics A⋅∇ takes on an important role as an operator in Hilbert space. As an ingredient in operators like the Hamiltonian or momentum, it allows formulation of physically valid equations. So while ∇⋅A and A⋅∇ may appear similar mathematically, their roles and interpretations differ significantly between classical and quantum domains.

Choosing the Appropriate Operator

To summarize, when working with vector operators involving del:

• ∇⋅A represents the divergence, a physically meaningful scalar quantity
• A⋅∇ is a differential operator with no direct physical meaning on its own
• In classical contexts like electromagnetism, only ∇⋅A is used in valid equations
• A⋅∇ becomes physically relevant as an operator ingredient in quantum mechanics So the appropriate operator choice depends on whether a classical or quantum scenario is being considered, and on whether a scalar physical quantity or operator differential is required.

  Higher Dimensional Generalizations

  The concepts of divergence and del operators can be readily generalized to higher dimensions. In n-dimensional euclidean space, the del operator is represented as: ∇ = î∂/∂x1 + ĵ∂/∂x2 + … + k̂n∂/∂xn
  And the divergence of a vector field A is given by: ∇⋅A = ∂A1/∂x1 + ∂A2/∂x2 + … + ∂An/∂xn So the physical and mathematical interpretations stay consistent when moving to higher dimensional settings.

  Conclusion

  In this article, I have aimed to distinguish between the ∇⋅A and A⋅∇ operators which, while related via dot products, represent fundamentally different physical quantities and roles depending on the classical or quantum mechanical context. A clear understanding of these subtleties is important when applying vector calculus concepts in physics.