Quantum Physics: Gaussian Wave Packets

In summary, the conversation is about a problem sheet and the solutions to two questions, 17 and 18. The focus is on showing that the probability density function is conserved and the integral of the wavefunction is equal to 1 for all t. The person working on the problem is not getting 1 and is asking for help. The solution is to plug in the expressions for r and θ to get the desired result.
  • #1
VVS
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Homework Statement


This is the problem sheet that I am solving at the moment:
View attachment T2SS14-Ex6.pdf


2. The attempt at a solution
I have already solved 17.
Here is my solution to 17:
View attachment Übung 17_3.pdf

Now I am working on 18.
I am trying to show that the probability density function is conserved. I.e the integral of the absolute value of the wavefunction in position space squared is equal to 1 for all t.
But somehow I am not getting 1.
Here is what I have so far:
View attachment Übung 18_3.pdf
 
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  • #2
You just have to plug your expressions for r and θ back in your result and you will get 1.
 
  • #3
Thanks you're right!
 

FAQ: Quantum Physics: Gaussian Wave Packets

1. What is a Gaussian wave packet in quantum physics?

A Gaussian wave packet in quantum physics is a wave function that represents a particle in a specific region of space with a specific range of momentum values. It is a superposition of many different plane waves, each with a different frequency and amplitude, but all centered around a central average value.

2. How is a Gaussian wave packet related to the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum. A Gaussian wave packet has a spread in both position and momentum, which means that it inherently obeys the Heisenberg uncertainty principle.

3. What is the Schrödinger equation and how does it relate to Gaussian wave packets?

The Schrödinger equation is the fundamental equation of quantum mechanics that describes the time evolution of a quantum system. Gaussian wave packets are often used as initial conditions for solving the Schrödinger equation, as they are a useful way to describe the state of a particle at a specific point in time.

4. How do Gaussian wave packets behave over time?

Gaussian wave packets typically spread out over time, meaning that the uncertainty in both position and momentum increases. This is due to the wave-like nature of particles in quantum mechanics, which allows them to exist in multiple states simultaneously and to spread out over time.

5. What are some real-world applications of Gaussian wave packets?

Gaussian wave packets have a wide range of applications in fields such as quantum computing, quantum cryptography, and quantum sensing. They are also used in modeling and simulating quantum systems, and in studying the behavior of particles in complex systems such as atomic nuclei.

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