Problem 8P

A definite-momentum wave function can be expressed by the formula Ψ (x) = A(cos kx +i sin kx), where A and k are constants.

(a) How is the constant k related to the particle’s momentum? (Justify your answer.)

(b) Show that, if a particle has such a wavefunction, you are equally likely to find it at any position x.

(c) Explain why the constant A must be infinitesimal, if this formula is to be valid for all x.

(d) Show that this wave function satisfies the differential equation dΨ/dx = ikΨ.

(e) Often the function cosθ + i sinθ is written instead as eiθ. Treating the i as an ordinary constant. show that the function. A eikx obeys the same differential equation as in part (d).

Problem 8P

Solution 8P

Step 1 :

Introduction :

In this question, we need to show a relation between constant k and momentum p

In the second part , we need to show, for the given wave function, the particle can be found at any x position

In the third part, we need to explain the reason for constant A to be infinitesimal for the formula to be valid for values of x

In the fourth part, we need to differentiate the given wave function to obtain

And finally, we need to show function obeys same differential equation and yields , if

is written as

Given wave function