Solving SE for Particle in Box: Energy Level Relation

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In summary, the conversation discussed solving the Schrodinger equation for a particle in a box with zero potential in one dimension. It was discovered that there are only certain permissible energy levels for the particle. The question was posed about the relationship between the number of boxes with a given energy when there are multiple boxes with a particle in each. It was concluded that the relationship would be arbitrary if there is no relationship between the boxes.
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Yajee
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I started learning Quantum mechanics(one dimension) a week ago and I came across solving the Schrodinger equation for a particle in a box with zero potential.I found out that there are only certain energy levels permissible for that particle. My query is if we take a large number of boxes with a particle in each, how will be the relation of number of boxes that have a given energy? Is that something governed by a law? Or does it depend on any physical condition? If so, what may it be?
 
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Hello Yayee, :welcome:

That would depend on whether there is any relationship between the boxes, I suppose. Can't think of anything physical.
 
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Okay, so if the boxes doesn't have any relationship, the relation(number of boxes having a given energy) can be arbitrary. Thank you!
 

1. How does the Schrödinger equation relate to the energy levels in a particle in a box?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system, such as a particle in a box. When solved for a particle in a box, the Schrödinger equation yields a set of energy eigenvalues, which represent the possible energy levels that the particle can have within the box.

2. What is the formula for the energy levels in a particle in a box?

The formula for the energy levels in a particle in a box is given by En = (n2h2)/8mL2, where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box.

3. How can the energy levels in a particle in a box be visualized?

The energy levels in a particle in a box can be visualized using a graphical representation known as an energy level diagram. In this diagram, the energy levels are shown as horizontal lines, with the lowest energy level at the bottom and the higher energy levels above it. The spacing between the energy levels increases as the quantum number n increases.

4. How does the size of the box affect the energy levels in a particle in a box?

The size of the box directly affects the energy levels in a particle in a box. As the length of the box increases, the spacing between the energy levels decreases, meaning that the energy levels become closer together. This is because the size of the box determines the allowed wavelengths of the particle, which in turn determines the possible energy levels.

5. What is the significance of the quantization of energy levels in a particle in a box?

The quantization of energy levels in a particle in a box is significant because it demonstrates the wave-like nature of particles at the quantum level. The fact that the energy levels can only take on discrete values rather than any value is a direct result of the wave-like behavior of particles. This quantization also has important implications for the behavior of matter at the microscopic level.

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