Infinite Square Well (Conceptual)

In summary, the conversation discusses the effect of a position measurement on a wave function defined as 1/sqrt(2)[ψ(1)+ψ(2)], where ψ represents the normalized stationary state energy eigenfunctions of the ISQ. It is suggested that the wave function would become a superposition of infinitely many stationary states, while still maintaining the conservation of energy. The time evolution operator is also mentioned as a means of determining the evolution of the wave function over time. It is clarified that the expectation value of energy would not remain the same before and after the measurement.
  • #1
Nicolaus
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Homework Statement


Say, for example, a wave function is defined as 1/sqrt(2)[ψ(1)+ψ(2)] where ψ are the normalized stationary state energy eigenfunctions of the ISQ.
Now, say I make a measurement of position. What becomes of the wavefunction at a time t>0 after the position measurement (i.e. after it has had time to evolve)?

The Attempt at a Solution


I would assume that the wave function would become a superposition of inifinitely many stationary states, but such that the expectation value of the Hamiltonian (energy) is the same so as to stay in line with the conservation of energy.
 
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  • #2
Nicolaus said:
I would assume that the wave function would become a superposition of inifinitely many stationary states, but such that the expectation value of the Hamiltonian (energy) is the same so as to stay in line with the conservation of energy.
That's why the time evolution operator commutes with Hamiltonian.
Nicolaus said:
What becomes of the wavefunction at a time t>0 after the position measurement (i.e. after it has had time to evolve)?
Let's say at t=0, you found your particle at ##x=x'## with probability ##|\langle x' | \Psi \rangle|^2##, then right after this point in time, the initial state before measurement collapses to ##|x' \rangle##. If you want to know how it evolves with time, then just apply the time evolution operator. If you want it to be expressed in the known entities such as the eigenstates of the ISW, insert the completeness operator in the space spanned by the ISW eigenstates in between the time evolution operator and ##|x' \rangle##.
 
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  • #3
Nicolaus said:
I would assume that the wave function would become a superposition of inifinitely many stationary states, but such that the expectation value of the Hamiltonian (energy) is the same so as to stay in line with the conservation of energy.
Do you mean that the expectation value of energy would be the same before and after the measurement? If so, that's not the case.
 
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Related to Infinite Square Well (Conceptual)

1. What is the concept of an infinite square well?

The infinite square well is a simplified model used in quantum mechanics to describe a particle confined to a one-dimensional region with infinite potential energy on either side. It serves as a useful concept to understand the behavior of particles in confined spaces.

2. How is the infinite square well potential defined mathematically?

The infinite square well potential is defined as a piecewise function, where the potential energy is zero within a finite region (the well) and infinite outside of it. Mathematically, it can be represented as V(x) = 0 for 0 < x < a and V(x) = ∞ for x < 0 and x > a, where a is the width of the well.

3. What is the significance of the energy levels in the infinite square well?

The energy levels in the infinite square well represent the allowed energy states for a particle confined in the well. These energy levels are discrete and evenly spaced, with the lowest energy level (n=1) being the ground state and the higher energy levels (n=2,3,4...) being excited states.

4. How do the energy levels change as the width of the well increases?

As the width of the well increases, the energy levels become closer together and the spacing between them decreases. This means that the energy required to move from one energy level to the next becomes smaller, and the particle becomes less confined within the well.

5. Can particles tunnel through an infinite square well?

No, particles cannot tunnel through an infinite square well because the potential energy is infinite outside of the well. Therefore, the wavefunction of the particle must be equal to zero at the boundaries, meaning that the particle cannot exist outside of the well.

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