# Particle in a Box: Probability of Ground State & Excited State

• jin85
In summary, a particle of mass m is initially confined to move in a box of length L with zero potential energy. The walls of the box are then pulled out to double the length, with the particle remaining in its initial state. After the walls are pulled out, the particle's position is measured and it is found to be at the center of the box, represented by a Dirac delta function. The energy of the particle is measured again. The question asks for the probability of the ground state or first excited state energy after this measurement. To solve this, you can use the wave function of the second potential (box of width 2L) written in terms of the energy eigenfunctions of the initial potential (box of width L).
jin85
Statement
A particle of mass m is confined to move in a box of length L. The potential energy of the particle within the box is zero and rises abruptly to a very large value (i.e. infinity) at the walls of the box. The walls of the box are now pulled out so that the box has length 2L. The walls are pulled out sufficiently quickly that instantaneously the state of the particle doesn’t change. The position of the particle is now measured and the outcome is that the particle is located exactly at the centre of the box (i.e. it is represented by a Dirac delta function located in the middle of the box). The energy of the particle is again measured.

Question
What is the probability that the outcome of this measurement will be the ground state energy? the first excited state?

My input
It actually contained more questions, but the last 2 are the ones i require help in. anyone with any idea? I am not too knowledgeable about dirac delta functions. I am not sure how i can get schrodinger's equation into something that can measure the probability of the ground state or 1st excited state.

Last edited:
Welcome to Physics Forums.

HINT: Can you write the wave function of the second potential (box of width 2L) in terms of the energy eigenfunctions of the initial potential (box of with L)?

## 1. What is a "Particle in a Box"?

A "Particle in a Box" is a theoretical model used in quantum mechanics to describe the behavior of a particle confined to a one-dimensional space. The particle is assumed to have no external forces acting on it and is confined by impenetrable walls on either end of the box.

## 2. What is the ground state and excited state of a particle in a box?

The ground state of a particle in a box is the lowest energy state that the particle can occupy. It is also known as the "n=1" state. The excited state, also known as the "n>1" state, is any state above the ground state where the particle has a higher amount of energy.

## 3. How is the probability of finding a particle in the ground state and excited state calculated?

The probability of finding a particle in a specific state is given by the square of the wave function at that state. For the ground state, the wave function is a constant value, resulting in a high probability of finding the particle in that state. For the excited states, the wave function has a more complex shape, resulting in a lower probability of finding the particle in those states.

## 4. What is the significance of the energy difference between the ground state and excited states?

The energy difference between the ground state and excited states is known as the "energy gap." This gap represents the minimum amount of energy required to excite the particle from the ground state to an excited state. It is an important factor in understanding the behavior of particles in a confined space.

## 5. How does the size of the box affect the probability of finding a particle in the ground state and excited state?

The size of the box has a direct impact on the probability of finding a particle in the ground state and excited state. As the size of the box decreases, the energy levels become more closely spaced, resulting in a higher probability of finding the particle in an excited state. Conversely, as the size of the box increases, the energy levels become more widely spaced, resulting in a higher probability of finding the particle in the ground state.

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