Question concerning particle in a box

In summary, the conversation discusses finding the probability density function for a particle in a box of width L, with an initial state of 1/sqrt(2) u1(x) + 1/sqrt(2) u2(x). The solution given is 1/2 u1(x) u1(x) + 1/2 u2(x) u2(x) + 2 cos (wt) u1(x) u2(x). The question is where the 2 cos (wt) comes from, which is explained by the fact that energy eigenstates evolve in time with a factor of exp(- i En t / hbar) associated with each state. When calculating the absolute square, the factor in question
  • #1
eku_girl83
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I need to find the probability density function for a particle in a box of width L. The initial state is 1/sqrt(2) u1(x) + 1/sqrt(2) u2(x), where u1(x) is the ground state and u2(x) is the first excited state. Omega is E2-E1/h

If I square this wave function, I get 1/2 u1(x)u1(x) +1/2 u2(x) u2(x) +u1(x)u2(x)

The solution given is 1/2 u1(x) u1(x) + 1/2 u2(x) u2(x) + 2 cos (wt) u1(x) u2(x)

My question is where does the 2 cos (wt) come from?
 
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  • #2
Remember that energy eigenstates evolve in time like exp(- i En t / hbar), so you have to associate such a factor with each of your states. If you now you calculate the absolute square, the factor in question should come right out.
 

1. What is a "particle in a box"?

A "particle in a box" is a theoretical model used to study the behavior of a particle confined to a one-dimensional space, such as a box. This model is often used in quantum mechanics to understand the properties of particles and their interactions within a confined space.

2. How does the size of the box affect the behavior of the particle?

The size of the box in the "particle in a box" model has a significant impact on the behavior of the particle. As the size of the box decreases, the energy levels of the particle become more discrete and the probability of finding the particle in certain areas of the box changes.

3. What is the significance of the "particle in a box" model in quantum mechanics?

The "particle in a box" model is used to illustrate important concepts in quantum mechanics, such as wave-particle duality and the quantization of energy. It also serves as a simplified representation of more complex systems and can be used to make predictions about the behavior of particles in other confined spaces.

4. Can the "particle in a box" model be applied to three-dimensional spaces?

While the "particle in a box" model is commonly used for one-dimensional spaces, it can also be extended to three-dimensional spaces. This is known as the "particle in a box in a box" model, where the particle is confined to a cube-shaped box. However, the calculations and equations become more complex in higher dimensions.

5. How does the uncertainty principle apply to the "particle in a box" model?

The uncertainty principle states that it is impossible to know both the exact position and momentum of a particle at the same time. In the "particle in a box" model, the confinement of the particle leads to a higher certainty in its position, but a lower certainty in its momentum. This concept is an important aspect of quantum mechanics and is evident in the behavior of particles in a confined space.

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