AP Physics 1 Exam Question: Quantum Theory

A particle of mass m is confined to a one-dimensional infinite potential well with width L. The particle is in its ground state.

a) Calculate the energy of the particle in terms of m, L, and the fundamental constant h. b) Find the most probable position for the particle within the well. c) What is the uncertainty in position of the particle?

Assume the following:

• The potential energy of the particle outside the well is infinity.
• The potential energy inside the well is zero.

Answer:

a) The energy of a particle in a one-dimensional infinite potential well can be calculated using the formula:

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

where

• E is the energy of the particle
• n is the quantum number (starting from 1)
• h is the Planck's constant (approximately 6.63 x 10^-34 J*s)
• m is the mass of the particle
• L is the width of the potential well

Since the particle is in its ground state, n = 1. Plugging these values into the formula, we get:

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

Simplifying further, we obtain:

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

b) The most probable position for the particle within the well can be found at the center of the well. Since the potential well is symmetric, the particle is equally likely to be found at any point along the center.

c) The uncertainty in position of the particle, represented by Δx, can be given by the formula:

Δx = L / (2√2)

where

• Δx is the uncertainty in position
• L is the width of the potential well

Plugging in the value of L into the formula, we have:

Δx = L / (2√2)

Simplifying further, we obtain:

Δx = L / (2.83)

Therefore, the uncertainty in position of the particle is L / (2.83).

Note: The values obtained are approximate values and may be rounded for ease of calculations.