Question

Question: Consider a cubic 3D infinite well of side length of L. There are 15 identical particles of mass m in the well, but for whatever reason, no more than two particles can have the same wave function. (a) What is the lowest possible total energy? (b) In this minimum total energy state, at what point(s) would the highest energy particle most likely be found? (Knowing no more than its energy, the highest energy particle might be in any of multiple wave functions open to it and with equal probability.)

Short answer

Answer

(a) The lowest possible total energy is $\frac{107{\pi }^2{h}^2}{2m{L}^2}$.

(b) The highest energy particle is most likely to found at the centre of the well.

Step-by-step solution

Identification of given data

The side length of 3D well is L.

The number of identical particles is n = 15 .

The mass of the identical particles is m.

The lowest energy wave function has $\left({\mathbf{n}}_{\mathbf{x}}\mathbf{,}{\mathbf{n}}_{\mathbf{y}}\mathbf{,}{\mathbf{n}}_{\mathbf{z}}\right)\mathbf{=}\left(\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\right)$. The next two lowest have (2,1,1), (1,2,1) and (1,1,2) . By combining each from these two there will six pairs for adding in existing two. Next higher will be (2,2,1) ,(2,1,2) and (1,2,2). This adds six to existing eight states. The fifteenth state is(3,1,1) ,(1,3,1) and (1,1,3) . for the last particle

Step 2(a): Determination of lowest possible total energy

The lowest possible total energy is given as:

$E_t=\left(\sum \left(n_x^2+n_y^2+n_z^2\right)\right)\left(\frac{{\pi }^2{h}^2}{2m{L}^2}\right)$

Here, h is the Planck’s constant.

Substitute all the values in the above equation.

$$\begin{gathered} E_t=\left[2\left(1^2+1^2+1^2\right)+6\left(2^2+1^2+1^2\right)+6\left(2^2+2^2+1^2\right)+\left(3^2+1^2+1^2\right)\right]\left(\frac{{\pi }^2{h}^2}{2m{L}^2}\right)\backslash \mathrm{hfill} \\ {E}_t=\frac{107{\pi }^2{h}^2}{2m{L}^2} \end{gathered}$$

Therefore, the lowest possible total energy is $\frac{107{\pi }^2{h}^2}{2m{L}^2}$.

Step 3(b): Determination of point for highest energy particles

Two among the three quantum numbers are unity, which gives maximum wave function at the centre of 3D well. The other quantum number is three for which there will be three maxima. This indicates that particle is most likely found at three points. The probability of finding particle is at the centre of the 3D well.

Therefore, the highest energy particle is most likely to found at the centre of the well.