Exploring the De Broglie Wave and Momentum in a Trapped Particle's Excited State

In summary, the de broglie wave for the 1st excited state of a particle trapped in a region of length L is a pure sine wave from 0 to 2pi, with a momentum of 2h/L according to the de broglie relation. However, according to the energy quantisation relation, the momentum for a state integer n=2 is actually h/L. This is because the particle-in-a-box eigenstates are not eigenstates of the momentum operator.
  • #1
Apashanka
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For a particle trapped in a region of length L the de broglie wave for the 1st excited state is a pure sine wave from 0 to 2pi
for which the particle momentum can be calculated as 2h/L from de broglie relation
Whereas from energy quantisation relation p=nh/2L where n is the state integer,for n=2 ,p=h/L
Why does so happen?
 
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  • #2
Apashanka said:
Whereas from energy quantisation relation p=nh/2L where n is the state integer,for n=2 ,p=h/L
Particle-in-a-box eigenstates are not eigenstates of the momentum operator, therefore you can't say that it corresponds to a given value of momentum.
 

1. What is the De Broglie wave and how does it relate to the momentum of a trapped particle in an excited state?

The De Broglie wave is a concept in quantum mechanics that describes the wave-like behavior of particles, which was first proposed by physicist Louis de Broglie in 1924. It suggests that all particles exhibit both wave-like and particle-like properties, and the wavelength of the De Broglie wave is inversely proportional to the momentum of the particle. In a trapped particle's excited state, the De Broglie wave can help us understand the distribution of momentum and energy within the system.

2. How is the De Broglie wave experimentally observed in a trapped particle's excited state?

The De Broglie wave can be observed through experiments such as electron diffraction and neutron interferometry. In these experiments, the particles are passed through a diffraction grating or an interferometer, respectively, which causes the particles to exhibit wave-like behavior and create a diffraction pattern or interference pattern. These patterns can then be analyzed to determine the wavelength and momentum of the particles, confirming the existence of the De Broglie wave.

3. What is the significance of studying the De Broglie wave and momentum in a trapped particle's excited state?

Studying the De Broglie wave and momentum in a trapped particle's excited state can provide valuable insights into the behavior of quantum systems and the fundamental principles of quantum mechanics. It can also have practical applications, such as in the development of quantum technologies, understanding chemical reactions, and studying the behavior of particles in extreme conditions.

4. Can the De Broglie wave and momentum of a trapped particle's excited state be controlled or manipulated?

Yes, the De Broglie wave and momentum can be controlled and manipulated in various ways, such as by changing the properties of the trapping potential, applying external fields or forces, or using quantum control techniques. These methods can be used to tailor the behavior of the particle and study its dynamics in a controlled manner.

5. Are there any limitations or challenges in studying the De Broglie wave and momentum in a trapped particle's excited state?

One of the main challenges in studying the De Broglie wave and momentum in a trapped particle's excited state is the delicate nature of quantum systems and the need for precise experimental techniques. The behavior of these systems is highly sensitive to external influences, and maintaining coherence in the particles can be challenging. Additionally, the interpretation of experimental results can also be complex, as quantum mechanics often defies our classical intuitions.

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