Ground state energy of 5 electrons in infinite well

[Saptarshi Sarkar]

Homework Statement

5 non-interacting electrons are placed in an infinite potential well of width a at T=0K. Calculate the maximum energy of the system.

Relevant Equations

$E_n = \frac {n^2\pi^2\hbar^2} {2ma^2}$

As the temperature given was 0K, I calculated the ground state energy of the system. I considered 2 electrons to be in the n=1 state, 2 in the n=2 state and 1 in the n=3 state by Pauli's exclusion principle.

By this configuration, I got the total energy of the system in the ground state to be

$E_{total} = 2\frac {1^2\pi^2\hbar^2} {2ma^2} + 2\frac {2^2\pi^2\hbar^2} {2ma^2} + \frac {3^2\pi^2\hbar^2} {2ma^2} = \frac {19\pi^2\hbar^2} {2ma^2}$

But, this doesn't match with any of the options provided in the question. What did I do wrong?

 

[mitochan]

Is $E_3$ included in the options? Your teacher may suggest "the maximum energy of the system" to be $E_3$, the highest energy level filled by electrons or Fermi energy, not E_total.

 

[PeroK]

Homework Statement:: 5 non-interacting electrons are placed in an infinite potential well of width a at T=0K. Calculate the maximum energy of the system.
Relevant Equations:: $E_n = \frac {n^2\pi^2\hbar^2} {2ma^2}$

As the temperature given was 0K, I calculated the ground state energy of the system. I considered 2 electrons to be in the n=1 state, 2 in the n=2 state and 1 in the n=3 state by Pauli's exclusion principle.

By this configuration, I got the total energy of the system in the ground state to be

$E_{total} = 2\frac {1^2\pi^2\hbar^2} {2ma^2} + 2\frac {2^2\pi^2\hbar^2} {2ma^2} + \frac {3^2\pi^2\hbar^2} {2ma^2} = \frac {19\pi^2\hbar^2} {2ma^2}$

But, this doesn't match with any of the options provided in the question. What did I do wrong?

This total energy is actually the minimum energy, given the exclusion principle.

 

[Saptarshi Sarkar]

This total energy is actually the minimum energy, given the exclusion principle.

But at temperature T=0K, shouldn't the energy be the minimum energy? I felt as if the maximum word was included to trick us.

Also, the maximum value as an option is $\frac {25\hbar^2\pi^2} {2ma^2}$

 

[PeroK]

But at temperature T=0K, shouldn't the energy be the minimum energy? I felt as if the maximum word was included to trick us.

Also, the maximum value as an option is $\frac {25\hbar^2\pi^2} {2ma^2}$

What are the options? I think we are (again) trying to guess what the question setter intended.

 

[Saptarshi Sarkar]

What are the options? I think we are (again) trying to guess what the question setter intended.

Currently can't provide the options as it was an online test and the the questions are not available yet. But from what I can remember, the available coefficients were 3,5,9 and 25.

 

[PeroK]

Currently can't provide the options as it was an online test and the the questions are not available yet. But from what I can remember, the available coefficients were 3,5,9 and 25.

Then, it's $9$ as @mitochan suggests.

 

[Dr Transport]

Key word here is non-interacting...

 

[PeroK]

Key word here is non-interacting...

Although that begs the question of whether non-interacting electrons have spin or not.

 

[Dr Transport]

Although that begs the question of whether non-interacting electrons have spin or not.

I'm thinking that non-interacting == ignore Fermi principle

 

[PeroK]

I'm thinking that non-interacting == ignore Fermi principle

Then they are all in the ground state and the maximum energy of the system is $5$ units. And, by maximum energy, the question setter means minimum energy? Or, should that be the maximum energy of anyone electron is $1$ unit?

It's very unclear to me what is meant by the question.

 

[Dr Transport]

Then they are all in the ground state and the maximum energy of the system is $5$ units. And, by maximum energy, the question setter means minimum energy? Or, should that be the maximum energy of anyone electron is $1$ unit?

It's very unclear to me what is meant by the question.

I agree, confusing wording.