The 1-D Infinite Potential Well

The 1-D infinite potential well is a simple model used in quantum mechanics to describe the behavior of a particle confined within a potential well of infinite height and finite width. In this model, the particle is assumed to be free to move within the well but cannot escape its boundaries.

The Wavefunction

The wavefunction of a particle in the 1-D infinite potential well is given by:

y(x) = Asin(πx/L)

where A is a constant and L is the width of the well. The wavefunction describes the probability distribution of finding the particle at a given position within the well.

Normalization

To determine the value of the constant A, we need to normalize the wavefunction. The normalization condition states that the integral of the absolute square of the wavefunction over all space must equal 1.

∫|y(x)|^2 dx = 1

In this case, we have:

∫|Asin(πx/L)|^2 dx = 1

Simplifying this integral gives:

|A|^2 ∫sin^2(πx/L) dx = 1

Using the trigonometric identity sin^2θ = (1 - cos2θ)/2, we have:

|A|^2 ∫(1 - cos(2πx/L))/2 dx = 1

Integrating each term separately, we get:

|A|^2 [(x - (L/2π)sin(2πx/L))/2] from 0 to L = 1

Evaluating the integral gives:

|A|^2 [(L - (L/2π)sin(2π)) - 0] = 2

Simplifying further, we find:

|A|^2 (L - (L/π)sin(2π)) = 2

Energy Levels

The possible values of energy for a particle in the 1-D infinite potential well are determined by the wavefunction. The energy of the particle is quantized and given by the equation:

E_n = (n^2π^2ħ^2)/(2mL^2)

where n is a positive integer, ħ is the reduced Planck's constant, and m is the mass of the particle.

Each energy level corresponds to a different value of n, and the energy levels are evenly spaced. The lowest energy level, n = 1, is called the ground state, and the higher energy levels are referred to as excited states.

Conclusion

In summary, the wavefunction of a particle in a 1-D infinite potential well is given by y(x) = Asin(πx/L). The constant A is determined by normalizing the wavefunction. The energy of the particle is quantized and given by the equation E_n = (n^2π^2ħ^2)/(2mL^2), where n is a positive integer. The possible values of energy for the particle are determined by the different energy levels, with the lowest energy level corresponding to the ground state.