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

[tanaygupta2000]

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.

 

[PeroK]

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 $E(1)$?

 

[tanaygupta2000]

How many electrons have you put in the ground state with $E(1)$?

Only 2 electrons can be accommodated at a time.

 

[PeroK]

Only 2 electrons can be accommodated at a time.

That isn't what I asked. I asked:

How many electrons have you put in the ground state with $E(1)$?

 

[tanaygupta2000]

That isn't what I asked. I asked:

In n=1, I put 2 electrons

 

[PeroK]

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?

 

[tanaygupta2000]

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

 

[PeroK]

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.

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.

 

[tanaygupta2000]

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.

 

[PeroK]

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?

 

[tanaygupta2000]

So, what is your answer for the total energy?

According to me it is 30h^2/8ma^2 which is 15h^2/4ma^2

 

[PeroK]

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:
$$E_{tot} = 2E(1) + 2E(2) + 2E(3) + 2E(4) = 2(1 + 4 + 9 + 16)E(1) = 60E(1)$$

 

[tanaygupta2000]

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)

 

[tanaygupta2000]

Sir I think it should be 30 E(1)View attachment 265312View attachment 265312

sorry for the double images

 

[PeroK]

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 $60E(1)$.

 

[tanaygupta2000]

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 $60E(1)$.

 

[PeroK]

Okay, so either the book's wrong or I'm wrong!

 

[DrClaude]

Okay, so either the book's wrong or I'm wrong!

I get the same thing as you. The book is wrong.

 

[Delta2]

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).

 

[PeroK]

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.

 

[Vanadium 50]

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.

 

[Delta2]

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.

 

[Vanadium 50]

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.

 

[Delta2]

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.

 

[Vanadium 50]

You are not going to convince me that this is a well-written question.

 

[PeroK]

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.

 

[Delta2]

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 .

 

[PeroK]

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!

 

[Delta2]

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...

 

[PeroK]

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

 

[Delta2]

... 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