Infinite Square Well and Energy Eigenstate question

In summary, A particle of mass m moving in the nth energy eigenstate of a one-dimensional infinite square well of width L has no uncertainty in its energy, as it is in a definite eigenstate. However, if the system is in a superposition of different eigenstates, there will be an uncertainty in its energy as it has a probability of having the energy of any of the component eigenstates. This is related to the postulate that only eigenvalues will be measured in any measurement of the observable associated with the operator Q.
  • #1
SeannyBoi71
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Hi all, just studying for my final exam and needed a little clarification on this.

Our prof did an example: Consider a particle of mass m moving in the nth energy eigenstate of a one-dimensional infinite square well of width L. What is the uncertainty in the particle's energy?

He said the answer was that there was no uncertainty simply because it was in this nth energy eigenstate. I don't really understand why this is the case. I thought maybe it had to do with this postulate he mentioned earlier: In any measurement of the observable associated with the operator Q, the only values that will ever be measured are the eigenvalues qn, which satisfy the eigenvalue equation Q*phi = q*phi. Can someone please help me with this?
 
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  • #2
Whenever a system is in an eigenstate of the Hamiltonian, there is a definite value of energy--that's the definition of an eigenstate. However, the system can also be in a superposition of different eigenstates, at which point there will be an uncertainty in the energy--it has a nonzero probability of having the energy of any of the component eigenstates. Does that make sense?
 

1. What is an infinite square well?

An infinite square well is a theoretical model used in quantum mechanics to describe the behavior of a particle confined within a one-dimensional space. It is represented by an infinitely high potential energy barrier on both sides, creating a well-like shape.

2. What are energy eigenstates in an infinite square well?

Energy eigenstates in an infinite square well refer to the specific energy levels that a particle can possess while confined within the well. These energy states are quantized, meaning they can only take on discrete values rather than a continuous range.

3. How are energy eigenstates related to the particle's position in the well?

The energy eigenstates are directly related to the probability of finding the particle at a specific position within the well. The higher the energy state, the greater the probability of finding the particle at the center of the well, and the lower the energy state, the greater the probability of finding the particle at the edges of the well.

4. How does the energy of a particle change in an infinite square well?

In an infinite square well, the energy of a particle can only change by discrete amounts, as it transitions from one energy eigenstate to another. When the particle is in a higher energy state, it has a greater potential energy and a smaller kinetic energy, and vice versa for a lower energy state.

5. What is the significance of the infinite square well in quantum mechanics?

The infinite square well is a fundamental model in quantum mechanics and is used to study the behavior of particles in confined spaces. It helps to illustrate the concept of quantization and the relationship between a particle's energy and its position. It also serves as a building block for more complex systems and is used in various applications, such as understanding the behavior of electrons in atoms.

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