Explain the physical origin of quantization for a particle confined to the interior of a one-dimensional box.

Question

Quizlet Admin · Accepted Answer

For explanation of physical origin of  $	ext{	extcolor{#c34632}{energy quantization}}$ of a particle confined in $	ext{	extcolor{#c34632}{one - dimensional box}}$, essential terms will be introduced.

The walls are impenetrable and potential energy tends to infinity (infinitely high walls assumed). Potential energy equals zero between the walls and it causes free - like behavior of the particle, possessing only kinetic energy. Therefore, particle is confined by walls (infinitely high potential energy for region between $x \leq 0$ and $x\geq L$.

Outside the box the value of wavefunction is zero, because it can not be found outside the box (high potential energy prevents that). These two conditions known as $	ext{	extcolor{#c34632}{boundary conditions}}$ are the main cause for the existence of quantized energy levels of free particle inside the box. This requirement implies that only some functions are possible and that is the reason why just specific energy levels are allowed. It can be shown mathematically if boundary conditions are applied to general solution of Schrodinger equation for a free particle. In other words to modify solution ($\psi = C sin kx + D cos kx$) in case of boundaries by including values of boundaries position into solution.
It follows:

at $\fbox{	extcolor{#4257b2}{	extcolor{#c34632}{$x = 0$}, $\psi = C sin 0 + D cos 0$  	extcolor{#c34632}{$\psi = D$}}}$

at $\fbox{	extcolor{#4257b2}{	extcolor{#c34632}{$x = L$}  $\psi = C sin kL + 0$  	extcolor{#c34632}{$\psi = C sin kL$}}}$

Related questions with answers

Explain the physical origin of quantization for a particle confined to the interior of a one-dimensional box.

Create an account to view solutions

Create an account to view solutions

Recommended textbook solutions

Pearson Chemistry

Chemistry: The Central Science

Atkins' Physical Chemistry

Modern Chemistry

More related questions