How Do Energy Levels Work for Electrons in a 1D Box?

In summary: Wait, I think I found my mistake. I was using 2 for ##n = 2## instead of 4.In summary, the total minimum energy for a system of eight electrons in a one dimensional box of length 'a' is found using Pauli's exclusion principle. This is calculated by placing two electrons in each of the four lowest energy levels (n=1,2,3,4) and using the formula E(n) = n^2 h^2/ (8mL^2). The total energy is given by 60E(1).
  • #1
tanaygupta2000
208
14
Homework Statement
Consider eight electrons in a one dimensional box of length 'a' extending from x = 0 to x = a. What is the minimum allowed total energy using Pauli's exclusion principle for the system (m = mass of electron)?
Relevant Equations
Energy of particles in a 1D box = n^2 h^2/ (8mL^2)
For the given problem, I know that the quantized energy for the particles in a 1D box is given by -
E(n) = n^2 h^2/ (8mL^2)

Here m = mass of electron
L = Length of the box = a

Now, since there are 8 electrons, but only 2 can occupy one energy level,
so I used n^2 = (1)^2 + (2)^2 = 1 + 4 = 5

So for a 'pair' of electrons, E = 5h^2/8ma^2
Hence total energy should be (since there are 8 electrons) = 4 * 5h^2/8ma^2
= 5h^2/2ma^2

Is my approach correct for attempting the question ?
Please guide.
 
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  • #2
tanaygupta2000 said:
Homework Statement:: Consider eight electrons in a one dimensional box of length 'a' extending from x = 0 to x = a. What is the minimum allowed total energy using Pauli's exclusion principle for the system (m = mass of electron)?
Relevant Equations:: Energy of particles in a 1D box = n^2 h^2/ (8mL^2)

For the given problem, I know that the quantized energy for the particles in a 1D box is given by -
E(n) = n^2 h^2/ (8mL^2)

Here m = mass of electron
L = Length of the box = a

Now, since there are 8 electrons, but only 2 can occupy one energy level,
so I used n^2 = (1)^2 + (2)^2 = 1 + 4 = 5

So for a 'pair' of electrons, E = 5h^2/8ma^2
Hence total energy should be (since there are 8 electrons) = 4 * 5h^2/8ma^2
= 5h^2/2ma^2

Is my approach correct for attempting the question ?
Please guide.

How many electrons have you put in the ground state with ##E(1)##?
 
  • #3
PeroK said:
How many electrons have you put in the ground state with ##E(1)##?
Only 2 electrons can be accommodated at a time.
 
  • #4
tanaygupta2000 said:
Only 2 electrons can be accommodated at a time.
That isn't what I asked. I asked:

PeroK said:
How many electrons have you put in the ground state with ##E(1)##?
 
  • #5
PeroK said:
That isn't what I asked. I asked:
In n=1, I put 2 electrons
 
  • #6
tanaygupta2000 said:
In n=1, I put 2 electrons
How many in ##n = 2##?

And, if your answer is ##2##, then where did you put the other four electrons?
 
  • #7
PeroK said:
How many in ##n = 2##?

And, if your answer is ##2##, then where did you put the other four electrons?
Sir I think I should do it like this:
Put 2 electrons in ---> n = 1
Put 2 electrons in ---> n = 2
Put 2 electrons in ---> n = 3
Put 2 electrons in ---> n = 4
(According to Pauli's exclusion principle, and since n = 1, 2, 3, 4 correspond to top 4 'minimum' energy levels)

So that E(1) = h^2/8ma^2
E(2) = 4 h^2/8ma^2
E(3) = 9 h^2/8ma^2
E(4) = 16 h^2/8ma^2

and therefore the total minimum energy is given by: E(1) + E(2) + E(3) + E(4) = 15 h^2/ 4ma^2
 
  • #8
tanaygupta2000 said:
For the given problem, I know that the quantized energy for the particles in a 1D box is given by -
E(n) = n^2 h^2/ (8mL^2)

Here m = mass of electron
L = Length of the box = a

First, this is not right.

tanaygupta2000 said:
and therefore the total minimum energy is given by: E(1) + E(2) + E(3) + E(4) = 15 h^2/ 4ma^2

You have eight electrons, not four.
 
  • #9
PeroK said:
First, this is not right.
You have eight electrons, not four.
I put 2 electrons in each energy level starting from n = 1 to n = 4 to accommodate all the eight electrons.
 
  • #10
tanaygupta2000 said:
I put 2 electrons in each energy level starting from n = 1 to n = 4 to accommodate all the eight electrons.
So, what is your answer for the total energy?
 
  • #11
PeroK said:
So, what is your answer for the total energy?
According to me it is 30h^2/8ma^2 which is 15h^2/4ma^2
 
  • #12
tanaygupta2000 said:
According to me it is 30h^2/8ma^2 which is 15h^2/4ma^2
Which is only wrong, I think, because you're using the wrong formula.

That said, really, you made this far more complicated than it need be:
$$E_{tot} = 2E(1) + 2E(2) + 2E(3) + 2E(4) = 2(1 + 4 + 9 + 16)E(1) = 60E(1)$$
 
  • #13
PeroK said:
Which is only wrong, I think, because you're using the wrong formula.

That said, really, you made this far more complicated than it need be:
$$E_{tot} = 2E(1) + 2E(2) + 2E(3) + 2E(4) = 2(1 + 4 + 9 + 16)E(1) = 60E(1)$$
Sir I think it should be 30 E(1)
abc.jpg
abc.jpg
 
  • #15
tanaygupta2000 said:
sorry for the double images
Okay, so you're using ##h## instead of ##\hbar##. So, you do have the right formula for ##E(1)##. Sorry about that.

Nevertheless, the total energy is ##60E(1)##.
 
  • #16
PeroK said:
Okay, so you're using ##h## instead of ##\hbar##. So, you do have the right formula for ##E(1)##. Sorry about that.

Nevertheless, the total energy is ##60E(1)##.
Capture.PNG
 
  • #17
Okay, so either the book's wrong or I'm wrong!
 
  • #18
PeroK said:
Okay, so either the book's wrong or I'm wrong!
I get the same thing as you. The book is wrong.
 
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  • #19
So two electrons that share the same energy state, have total energy 2 times the energy of the state E. What if someone argues that the total energy is E because it is shared by the two electrons (have in mind that i am absolute beginner in quantum mechanics).
 
  • #20
Delta2 said:
So two electrons that share the same energy state, have total energy 2 times the energy of the state E. What if someone argues that the total energy is E because it is shared by the two electrons (have in mind that i am absolute beginner in quantum mechanics).
Two particles in an energy state both have the energy of that state. That's what it means.
 
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  • #21
60/8 is 15/2. No matter who says otherwise.

This is just a bad question. A student might not know that they want her or him to ignore the electrostatic interaction between electrons.
 
  • #22
Vanadium 50 said:
60/8 is 15/2. No matter who says otherwise.

This is just a bad question. A student might not know that they want her or him to ignore the electrostatic interaction between electrons.
I haven't done exact calculations but I think the electrostatic potential energy of two electrons must be really small in comparison with the ##E(1)## energy here.
 
  • #23
Delta2 said:
I haven't done exact calculations

That is unnecessary, but you probably should have thought about it a bit more. Inter-electron potential goes as 1/a, and energies in a square well go as 1/a2. Therefore, for some lengths it matters and for some it doesn't.

Including this turns a simple problem into an impossible one. Those of us with experience in QM know this, but one shouldn't expect a rank beginner to know "what the problem author really meant." It's a poorly written problem.

And 60/8 is still 15/2.
 
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  • #24
Vanadium 50 said:
That is unnecessary, but you probably should have thought about it a bit more. Inter-electron potential goes as 1/a, and energies in a square well go as 1/a2. Therefore, for some lengths it matters and for some it doesn't.
Well i did some calculations after all and it turns out that the two energies are comparable. Though it seems that the reason you mention (1/a vs 1/a^2) is not the main reason for this, the main reason is that plank's constant appears squared in the quantized energy levels of the particle in the box(something i didnt notice earlier) , and that makes E(1) really small too.
 
  • #25
You are not going to convince me that this is a well-written question.
 
  • #26
Delta2 said:
Well i did some calculations after all and it turns out that the two energies are comparable. Though it seems that the reason you mention (1/a vs 1/a^2) is not the main reason for this, the main reason is that plank's constant appears squared in the quantized energy levels of the particle in the box(something i didnt notice earlier) , and that makes E(1) really small too.
The question is only really valid for non-interacting fermions. Having charged particles requires a very different Hamiltonian and energy levels.

Personally, I think texts should be honest when they are using a "toy" theory. In this case we can't really be dealing with electrons.
 
  • #27
I think this question is written ok. Also I believe the book considers the total energy of the different states used and not the total energy of the particles. Which ok you are right that it is wrong, states are abstract ideas and don't possesses real energy, particles possesses real energy .
 
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  • #28
Delta2 said:
I think this question is written ok. Also I believe the book considers the total energy of the different states used and not the total energy of the particles. Which ok you are right that it is wrong, states are abstract ideas and don't possesses real energy, particles possesses real energy .
That's nonsense!
 
  • #29
PeroK said:
That's nonsense!
What's exactly nonsense, I am saying that particles posses specific energy when they are in specific states. Its not the states that have the energy...
 
  • #30
Delta2 said:
What's exactly nonsense, I am saying that particles posses specific energy when they are in specific states. Its not the states that have the energy...
... particles are abstract ideas! Particles are states!

You should read my beginner's guide to baryons :wink:
 
Last edited:
  • #31
PeroK said:
... particles are abstract ideas! Particles are states!
I am reading it now...yes well according to modern quantum field theory particles are states of the underlying field
 

Related to How Do Energy Levels Work for Electrons in a 1D Box?

1. What is a 1D box in relation to electrons?

A 1D box is a theoretical model used to describe the behavior of electrons in a confined space, such as a nanoscale wire or semiconductor device. It assumes that the electrons are confined to move only in one dimension, along a line or axis.

2. How does the size of the 1D box affect the behavior of electrons?

The size of the 1D box has a significant impact on the behavior of electrons. As the box size decreases, the energy levels of the electrons become more discrete and the energy gap between levels increases. This results in a higher energy barrier for the electrons to overcome, leading to a decrease in their mobility and conductivity.

3. What is the significance of the quantum confinement effect in a 1D box?

The quantum confinement effect refers to the phenomenon where the behavior of electrons is altered due to their confinement in a small space. In a 1D box, this leads to a quantization of energy levels, resulting in discrete energy states and a decrease in the number of available states for electrons to occupy. This effect is essential in understanding the behavior of electrons in nanoscale devices.

4. How does the potential energy barrier in a 1D box affect the movement of electrons?

The potential energy barrier in a 1D box acts as a barrier for the movement of electrons, as they can only occupy discrete energy levels within the box. As the barrier height increases, the energy levels become more discrete, and the electrons are less likely to overcome the barrier and move to a different energy state. This results in a decrease in the electron's mobility and conductivity.

5. Can the behavior of electrons in a 1D box be described by classical physics?

No, the behavior of electrons in a 1D box cannot be accurately described by classical physics. Classical physics assumes that particles can have any energy value, whereas in a 1D box, the energy levels are discrete and quantized. The behavior of electrons in a 1D box can only be accurately described by quantum mechanics.

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