Is My Approach to the Gaussian Wave Packet Problem Professional?

In summary, the discussion evaluates the professionalism of an individual's approach to the Gaussian wave packet problem, focusing on methodology, clarity, and adherence to established principles in quantum mechanics. It emphasizes the importance of rigor in mathematical formulations and the significance of clear communication of concepts to enhance understanding and collaboration in the field.
  • #1
BlondEgg
4
1
Homework Statement
Problem involving normalization and Fourier transform
Relevant Equations
Wave function
hi,

I'm solving this statement,
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1721513995402.png

We split into parts
1721514094971.png

1721514116323.png

1721514143924.png

Can some expert in Quantum say that my working is professional?
Kind wishes to you
 
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  • #2
Parts (i) and (ii) are rigorously explained so one may say that they are professionally done. Part (iii) is not "professional", mainly because you make an assertion without justifying it. You say that "the probability density ##~| \Psi(x,t)|^2~## spreads over time indicating that the wave packet disperses as time progresses." To make the argument stick, you need to do the integral in equation (13), find ##~|\Psi(x,t)|^2~##, identify the width and argue that it is time-dependent.
 
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Likes Greg Bernhardt

FAQ: Is My Approach to the Gaussian Wave Packet Problem Professional?

1. What is a Gaussian wave packet?

A Gaussian wave packet is a specific type of wave function that describes a localized particle in quantum mechanics. It is characterized by a Gaussian shape in position space, which means it has a peak at a certain position and decays smoothly to zero as you move away from that position. This type of wave packet is often used because it minimizes the uncertainty in both position and momentum, making it a useful tool for studying quantum systems.

2. How do I construct a Gaussian wave packet?

To construct a Gaussian wave packet, you start with a Gaussian function in position space, which can be mathematically expressed as: Ψ(x) = A * exp(-(x - x0)² / (2σ²)) * exp(ikx), where A is the normalization constant, x0 is the central position, σ is the width of the packet, and k is the wave number related to the momentum. You then ensure that the wave packet is properly normalized so that the total probability of finding the particle is one.

3. What are the key properties of Gaussian wave packets?

Gaussian wave packets have several key properties, including their ability to maintain their shape while propagating through space, which is a result of the uncertainty principle. They also exhibit minimal dispersion, meaning that the spread of the wave packet over time is slower compared to other forms of wave packets. Additionally, Gaussian wave packets can be described entirely by their mean position and momentum, making them mathematically convenient for analysis.

4. How does the width of a Gaussian wave packet affect its behavior?

The width of a Gaussian wave packet, represented by σ, directly affects its localization and momentum uncertainty. A narrower wave packet (smaller σ) is more localized in position space, which results in a greater uncertainty in momentum, according to the Heisenberg uncertainty principle. Conversely, a wider wave packet (larger σ) is less localized but has a smaller momentum uncertainty. This trade-off is crucial for understanding the behavior of quantum particles.

5. What applications do Gaussian wave packets have in physics?

Gaussian wave packets are widely used in various fields of physics, particularly in quantum mechanics and quantum optics. They serve as models for studying the dynamics of quantum particles, analyzing scattering processes, and exploring wave-particle duality. Additionally, Gaussian wave packets are used in the development of quantum information technologies, such as quantum computing and quantum communication, due to their mathematical simplicity and physical relevance.

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