Identify which of the following function are Eigen functions of the operators d/dx and give the corresponding Eigen value
(1). ¥= e^ikx
(2). ¥= coskx
(3). kx
To determine if a function is an eigenfunction of an operator, we need to check if applying the operator to the function results in a constant multiple of the same function.
Let's start with the first function, ψ = e^(ikx), and see if it is an eigenfunction of the first derivative operator, d/dx.
We apply the first derivative operator to ψ:
d/dx [e^(ikx)] = ik * e^(ikx)
We can see that applying the operator to ψ does not result in a constant multiple of ψ. Instead, we have an additional factor of ik. Therefore, ψ = e^(ikx) is not an eigenfunction of the first derivative operator.
Now, let's move on to the second function, ψ = cos(kx), and check if it is an eigenfunction of the first derivative operator.
We apply the first derivative operator to ψ:
d/dx [cos(kx)] = -k * sin(kx)
We can see that applying the operator to ψ does not result in a constant multiple of ψ. Instead, we have an additional factor of -k. Therefore, ψ = cos(kx) is not an eigenfunction of the first derivative operator.
Finally, let's examine the third function, ψ = kx, and determine if it is an eigenfunction of the first derivative operator.
We apply the first derivative operator to ψ:
d/dx [kx] = k
We can see that applying the operator to ψ results in a constant multiple of ψ. Therefore, ψ = kx is an eigenfunction of the first derivative operator, with an eigenvalue of k.
To summarize:
1. ψ = e^(ikx) is not an eigenfunction of d/dx.
2. ψ = cos(kx) is not an eigenfunction of d/dx.
3. ψ = kx is an eigenfunction of d/dx, with an eigenvalue of k.