AP Physics 2 Exam Question:

Consider a particle in a one-dimensional infinite square well potential. The width of the well is given by "L" units.

a) Sketch the wavefunction for the particle in the ground state. Label the important features of the wavefunction.

b) What is the energy of the particle in the ground state of the infinite square well potential, in terms of the width "L" and fundamental constants?

c) Assuming the particle is in the ground state, calculate the probability of finding it in the left half of the well.

d) Suppose the width "L" of the well is doubled. How will this change affect the energy of the particle in the ground state?

e) Explain, using the Uncertainty Principle, why a particle in the ground state cannot have a definite position and momentum simultaneously.

Answer:

a) The wavefunction for the particle in the ground state of a one-dimensional infinite square well potential is given by:

The important features of the wavefunction are as follows:

• The wavefunction is symmetric about the center of the well.
• The wavefunction approaches zero at both ends of the well.
• The wavefunction has a node at the center of the well.

b) The energy of the particle in the ground state of an infinite square well potential is given by the equation:

Where "n" is the energy level (for the ground state, n = 1), "h" is Planck's constant (6.626 x 10^(-34) J·s), and "m" is the mass of the particle.

Therefore, the energy of the particle in the ground state is:

E = (1^(2) * h^(2)) / (8 * m * L^(2))

c) The probability of finding the particle in a specific region is given by the square of its wavefunction in that region. In this case, we need to calculate the squared wavefunction in the left half of the well.

Since the wavefunction is symmetric about the center, the probability of finding the particle in the left half of the well is equal to the probability of finding it in the right half.

Thus, the probability of finding the particle in the left half of the well is 0.5, or 50%.

d) Doubling the width "L" of the well will decrease the energy of the particle in the ground state.

From the energy equation, we can see that the energy is inversely proportional to the square of the width "L". Therefore, if the width is doubled, the energy will decrease by a factor of 4.

e) According to the Uncertainty Principle, there is a fundamental limit to the precision with which certain pairs of physical properties can be known simultaneously.

In the case of position and momentum, the Uncertainty Principle states that the product of the uncertainties in position (Δx) and momentum (Δp) must be greater than or equal to Planck's constant divided by 2π:

Δx * Δp ≥ h / (2π)

In the ground state, the wavefunction is localized within a finite region, leading to a small uncertainty in position (Δx). However, due to the wave nature of particles, the uncertainty in momentum (Δp) becomes large.

Therefore, it is not possible for a particle in the ground state to have a definite position and momentum simultaneously, as the Uncertainty Principle imposes limits on the precision with which these properties can be simultaneously known.