Ground state energy of 5 electrons in infinite well

In summary: But based on the options given and the fact that it's non-interacting electrons, I would say the maximum energy of the system is ##9## units. In summary, the question involves calculating the maximum energy of a system of 5 non-interacting electrons placed in an infinite potential well at T=0K, with options for the coefficient being 3, 5, 9, and 25. However, due to the exclusion principle, the total energy calculated for the ground state is actually the minimum energy. Therefore, the maximum energy of the system is ##9## units.
  • #1
Saptarshi Sarkar
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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?
 
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  • #2
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.
 
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  • #3
Saptarshi Sarkar said:
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.
 
  • #4
PeroK said:
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}##
 
  • #5
Saptarshi Sarkar said:
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.
 
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  • #6
PeroK said:
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.
 
  • #7
Saptarshi Sarkar said:
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.
 
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  • #9
Dr Transport said:
Key word here is non-interacting...
Although that begs the question of whether non-interacting electrons have spin or not.
 
  • #10
PeroK said:
Although that begs the question of whether non-interacting electrons have spin or not.
I'm thinking that non-interacting == ignore Fermi principle
 
  • #11
Dr Transport said:
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.
 
  • #12
PeroK said:
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.
 

Related to Ground state energy of 5 electrons in infinite well

1. What is the ground state energy of 5 electrons in an infinite well?

The ground state energy of 5 electrons in an infinite well depends on the specific dimensions of the well and the energy levels of the individual electrons. It can be calculated using the Schrodinger equation and the principles of quantum mechanics.

2. How does the ground state energy of 5 electrons in an infinite well compare to other energy levels?

The ground state energy is the lowest possible energy level for a system of particles in an infinite well. It is lower than the first excited state and all subsequent energy levels.

3. Can the ground state energy of 5 electrons in an infinite well be negative?

No, the ground state energy cannot be negative. It represents the lowest possible energy state of a system and must be greater than or equal to zero.

4. How does the ground state energy of 5 electrons in an infinite well change with different well dimensions?

The ground state energy is directly proportional to the size of the well. As the well dimensions increase, the ground state energy also increases.

5. What is the significance of the ground state energy in quantum mechanics?

The ground state energy is important in quantum mechanics because it represents the lowest possible energy state of a system. It serves as a reference point for calculating and understanding the energy levels and behavior of particles in a given system.

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