Question

Show that the uncertainty in a particle's position in an infinite well in the general case of arbitrary n is given by

√ 1/12 – 1/2n² π²
Discuss the dependence. In what circumstance does it agree with classical uncertainty of L/ √12 iscuss in excericise 55?

Answer 1

The uncertainty in a particle's position in a general case infinite well depends on the quantum number, with higher values of n resulting in lower uncertainty.

Step-by-step explanation:

The uncertainty in a particle's position in an infinite well in the general case of arbitrary n is given by √(1/12 – 1/2n²) π². This equation describes the dependence of uncertainty in position on the quantum number, n. As n increases, the uncertainty in position decreases. This means that for higher energy levels, the particle's position can be determined more accurately.

In the special case of n=1, the uncertainty in position is √(1/12) π², which is the same as the classical uncertainty of L/√12 for a particle in a one-dimensional box. This occurs because for the ground state (n=1), the particle is confined to the smallest possible region within the well, and its uncertainty in position is equal to the width of the well, L.

However, for n=1, the uncertainty is the same as the classical uncertainty for a particle in a one-dimensional box, indicating that the particle is confined to the smallest possible region within the well.