In quantum mechanics, the derivative operator d/dx is a linear operator. To prove this, we need to demonstrate that it satisfies the properties of linearity: additivity and homogeneity.
Additivity: Let's consider two arbitrary functions f(x) and g(x), and their derivatives with respect to x, denoted as f'(x) and g'(x) respectively. We want to show that the derivative operator satisfies the property of additivity, which means that the derivative of the sum of two functions is equal to the sum of their derivatives.
Mathematically, we have: d/dx [f(x) + g(x)] = f'(x) + g'(x)
To prove this, we can apply the definition of the derivative:
d/dx [f(x) + g(x)] = lim(h->0) [(f(x+h) + g(x+h) - f(x) - g(x))/h]
By expanding and rearranging the terms, we get:
= lim(h->0) [f(x+h) - f(x)]/h + lim(h->0) [g(x+h) - g(x)]/h
This is equivalent to f'(x) + g'(x), which demonstrates the additivity property of the derivative operator.
Homogeneity: The derivative operator also satisfies the property of homogeneity, which means that scaling a function by a constant scales its derivative by the same constant. Let's consider an arbitrary function f(x) and a constant c.
We want to show that: d/dx [c * f(x)] = c * f'(x)
Applying the definition of the derivative:
d/dx [c * f(x)] = lim(h->0) [(c * f(x+h) - c * f(x))/h]
By factoring out the constant c, we have:
= c * [lim(h->0) (f(x+h) - f(x))/h]
This is equal to c * f'(x), which verifies the homogeneity property of the derivative operator.
By demonstrating both the additivity and homogeneity properties, we have proven that the derivative operator d/dx is a linear operator in quantum mechanics.