Question

show that the commutator (x,d/dx)=-1

Answer 1

Operators

An operator is a symbol which defines the mathematical

operation to be cartried out on a function.

Examples of operators:

d/dx = first derivative with respect to x

√ = take the square root of

3 = multiply by 3

Operations with operators:

If A & B are operators & f is a function, then

(A + B) f = Af + Bf

A = d/dx, B = 3, f = f = x2

(d/dx +3) x2 = dx2

/dx +3x2 = 2x + 3 x2

ABf = A (Bf)

d/dx (3 x2

) = 6x

Note that A(Bf) is not necessarily equal to B(Af):

A = d/dx, B = x, f = x2

A (Bf) = d/dx(x⋅ x2

) = d/dx (x3

) = 3 x2

Answer 2

Given: Given commutator is (x, d/dx).

To find: We have to prove that its value is -1.

Solution:

We can determine the value of the commutator by the following steps-

The commutator is [x, d/dx].

Let the wave function operate on the commutator is w.

So, the commutator can be written as-

[x, d/dx]w

$(x. \frac{d}{dx} w - \frac{d}{dx}xw) = (x.\frac{d}{dx})w \\ (x\frac{dw}{dx} - x\frac{dw}{dx} - w\frac{dx}{dx}) = (x.\frac{d}{dx})w \\ - w = (x.\frac{d}{dx})w \\ (x.\frac{d}{dx}) = - 1$

Thus, the value of the commutator is -1.

So, x and d/dx does not commute as their value is -1. If the value of the commutator is 0 then they commute.

So, the commutator (x, d/dx)=-1 (proved).