Question

(Q)Consider a one-dimensional box extends from x = x0 to x = x0 + L, where x0 6= 0.

Would the expression for the wave function of a particle in this box be any different

from those in the box that extends from x = 0 to x = L? Would the energy levels be

different?

Answer 1

The expression for the wavefunction for the box that  extends from x = 0 to x = L is given as follows (see https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Mechanics/05.5%3A_Particle_in_Boxes/Particle_in_a_1-Dimensional_box):

$$\psi=\sqrt{\dfrac{2}{L}}\sin\dfrac{n\pi}{L}x$$

Since this expression depends on $x$, then the coordinate shift $x\to x-x_0$ wich correspons to the  one-dimensional box extends from x = x0 to x = x0 + L will lead to the following expression:

$$\psi=\sqrt{\dfrac{2}{L}}\sin\dfrac{n\pi}{L}(x-x_0)$$

Thus, the wavefunction has changed.

On the other hand, the expression for energy levels depends only on $L$ (the width of the box):

$$E_n = \dfrac{n^2h^2}{8mL^2}$$

thus, remains unchanged under the box shift.