# Measurement for a particle in a box

In summary, the particle in a one dimensional box with walls at (-a/2, +a/2) has energy probabilities P(E1) = 1/3, P(E2) = 1/3, P(E,3) = 1/3, P(En) = 0 for all other n. If the parity of the state is measured and +1 is found, the wavefunction must be in an eigenstate of the parity operator with eigenvalue +1. The parity operator and Hamiltonian commute, so the wavefunction is in a linear combination of eigenstates for n = 1 and n = 3, since those were the original even eigenfunctions used in the expression of the wavefunction

It is known that a particle in a one dimensional box with walls at (-a/2, +a/2) has energy probabilities:
P(E1) = 1/3, P(E2) = 1/3, P(E,3) = 1/3, P(En) = 0 for all other n. If the parity of the state is measured and +1 is found, what can you say about the value of the measurement of E sometime later?

The eigenstates of the particle in a box are root(2/a) sin(n*pi*x/a) for even n and root(2/a) cos(n*pi*x/a) for odd n.

So I wrote down the initial state as a linear combination of eigenstates of the Hamilitonian corresponding to n = 1, 2, 3 each with coefficient 1/root3. Now, since the measured parity was +1, the wavefunciton must be in an eigenstate of the parity operator with eigenvalue +1. The parity operator and Hamiltonian commute for this system so I can use the Hamiltonian eigenstates. My question is, does the wavefunction now have to be in a linear combination of eigenstates for n = 1 and n = 3 since those were the original even eigenfunctions used in the expression of the wavefunction? Or is it in a linear combination of all even eigenstates of the Hamiltonian?

If |psi> is the state before the parity measurement, then immediately after the parity measurement, the state of system is the (normalized) projection of |psi> into the subspace of all even parity states.

What is this projection?

## 1. How is the energy of a particle in a box measured?

The energy of a particle in a box is measured using the Schrödinger equation, which calculates the energy levels of the particle based on its mass, the length of the box, and the potential energy within the box.

## 2. What is the significance of a particle in a box in quantum mechanics?

A particle in a box is a common model used in quantum mechanics to understand the behavior of particles confined to a limited space. It helps to illustrate the concepts of quantization and the wave-particle duality of matter.

## 3. How does the size of the box affect the energy of the particle?

The size of the box has a direct effect on the energy of the particle. As the box size decreases, the energy levels become more closely spaced, and the energy of the particle increases. This is because the particle is more confined and has less space to move in, leading to higher energy levels.

## 4. Can a particle in a box have a negative energy?

No, a particle in a box cannot have a negative energy. According to the Schrödinger equation, the energy levels of a particle in a box are always positive. However, the particle can have a negative potential energy if it is in a region with a negative potential, such as a potential well.

## 5. What is the probability of finding a particle in a certain energy state in a box?

The probability of finding a particle in a certain energy state is proportional to the square of the amplitude of the corresponding wavefunction. This means that the higher the energy level, the lower the probability of finding the particle in that state, and vice versa.