Quantum Mechanics - Gaussian Wave Packet

A Gaussian wave packet travels through free space. Which of the following statement about the wave packet are correct for all such wave packets?

1. The average momentum of the wave packet is zero
2. The width of the wave packet increases with time, as t → ∞.
3. The amplitude of the wave packet remains constant with time.
4. The narrower the wave packet is in momentum space, the wider it is in coordinate space.

A. I and III only
B. II and IV only
C. I, II, and IV only
D. II,III, and IV only
E. I, II,III, and IV only

(GR9677 #76)

Solution:

The Gaussian wave packet satisfies the Heisenberg uncertainty principle: ΔxΔp ≥ ħ/2

→ Statement IV is TRUE.
→ Statement I is FALSE, Δp cannot be 0
→ Answer A, C, E are FALSE.

Answer D suggests that both II and III are true.
If II and III are true, the area of gaussian wave packet can go to ∞
This is not allowed by the Heisenberg uncertainty principle.
→ D is FALSE

Answer: B

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

Lab Methods - Standard Deviation

The magnitude of the force F on an object can be determined by measuring both the mass m of an object and the magnitude of its acceleration a, where F = ma. Assume that these measurements are uncorrelated and normally distributed. if the standard deviations of the measurements of the mass and acceleration are σ_m and σ_a respectively, then σ_F/F is

A.

B.

C.

D.

E.

(GR9277 #48)

Solution:

Multiplying two quantities with uncertainty:

u = x ⋅ y



For F = ma:



Answer: C

Lab Methods - Uncertainty

A student makes 10 one-second measurement of the disintegration of a sample of a long lived radioactive isotope and obtains a following values: 3, 0, 2, 1, 2, 4, 0, 1, 2, 5. How long should the student count to establish the rate to an uncertainty of 1 percent?

A. 80 s
B. 160 s
C. 2000 s
D. 5000 s
E. 6400 s

(GR0177 #16)

Solution:

Radioactive decay can be described by Poisson Distribution.

Poisson Distribution (PD):
Probability distribution of discrete events over an interval (time. distance, etc)

In PD, Standard Deviation, σ = √μ  (see problem GR8677 #40)
μ = λT = expected value
λ = average rate
T = time interval

% Uncertainty = (σ/μ) × 100%

σ/μ = 0.01
√μ/μ = 10⁻²
μ/μ² = 10⁻⁴
1/μ = 1/10⁴
μ = λT = 10⁴

λ = (3 + 0 + 2 + 1 + 2 + 4 + 0 + 1 + 2 + 5)/10 = 2
T = 10⁴/2 = 5000

Answer: D