Quantum Mechanics - Wave Function

If a freely moving electron is localized in space to within ∆x₀ of x₀, its wave function can be described by a wave packet

where f(k) is peaked around a central value k₀. Which of the following is most nearly the width of the peak in k?

A. 

B.

C.

D.

E.

(GR9277 #27)

Solution:

In quantum mechanics, the momentum p = ħk and position x wave functions are Fourier transform pairs and the relation between p and x representations forms the Heisenberg uncertainty relation:

∆x∆k ≥ 1 → ∆k ≥ 1/∆x

Or, since k and x are fourier variables, their localization would vary inversely.

Answer: B

Electromagnetism - Fourier Series

If n is an integer ranging from 1 to infinity, is an angular frequency, and t is time, then the Fourier series for a square wave, as shown above, is given by which of the following?

A.

B. 

C. 

D. 

E.  

(GR9277 #39)

Solution:

The sine function is an odd function.
The cosine function is an even function.
The function in this problem is an odd square wave function.
→ C, D, E are FALSE.

Both A and B are the same, thus the right solutions. However, A is more generalized and B is the simplified answer.

For Answer A, the sine term vanishes when n is even (n = 0, 2, 4, ...) → rewrite n for odd term as 2n +1 with n starts at 0 → the answer can be simplified to B.

Answer: B