Thought experiment about wave functions

[ftr]

Suppose we have a particle, let's say an electron, in a box of size D. And we add another one next to it at some distance L center to center. How do we solve for the wavefunctions of the electron. Can it be solved in normal QM or do we need QFT. Thanks.

 

[BvU]

Hi,

WHat would be the difference between the Hamiltonian for a single electron and one for two electrons in a box ?

 

[Nugatory]

As @BvU's answer suggests, you do not need QFT for this problem; ordinary garden-variety non-relativistic QM is sufficient.

But one more hint might be helpful: you are not solving for the wavefunctionS (plural) of two electrons. You are solving for the one wavefunction of the one quantum system which consists of two electrons.

 

[ftr]

Hi,

WHat would be the difference between the Hamiltonian for a single electron and one for two electrons in a box ?

the setup I am referring to is two boxes(at distance L) with each having an electron, which means they should repel each other.

 

[Nugatory]

the setup I am referring to is two boxes(at distance L) with each having an electron, which means they should repel each other.

Right... so what's the Hamiltonian for that system?

 

[ftr]

Right... so what's the Hamiltonian for that system?

I don't know, I was hoping somebody would answer that.

 

[BvU]

You can do better than that: you mention two electrons, not two neutrons or an electron and a posittron !

 

[HAYAO]

I don't know, I was hoping somebody would answer that.

You might want to look up what Hamiltonian is then. You know that Hamiltonian consists of kinetic and potential energy terms, right? If you want to make constraints that each of the two electrons are inside two separate potential box, what do you need to do to the potential energy terms?

 

[ftr]

You might want to look up what Hamiltonian is then. You know that Hamiltonian consists of kinetic and potential energy terms, right? If you want to make constraints that each of the two electrons are inside two separate potential box, what do you need to do to the potential energy terms?

Of course I know about the standard particle in a box treatment with the Hamiltonian. But here I have two of them interacting, I know about the Zee QFT treatment but I want to know If an expression for the wavefunction can be derived. This is not a homework problem it is just a thought experiment. Thank you all for your help.

 

[HAYAO]

Of course I know about the standard particle in a box treatment with the Hamiltonian. But here I have two of them interacting, I know about the Zee QFT treatment but I want to know If an expression for the wavefunction can be derived. This is not a homework problem it is just a thought experiment. Thank you all for your help.

You don't need QFT for this, and this is not a thought experiment neither.

You have two potential boxes. You have two electrons. Potential energy must be accounted for both. How would you express the potential energy of two electrons interacting? (This is almost like an answer, in fact you said yourself. You are very close to achieving Hamlitonian for the problem you are looking at!)

 

[ZapperZ]

Of course I know about the standard particle in a box treatment with the Hamiltonian. But here I have two of them interacting, I know about the Zee QFT treatment but I want to know If an expression for the wavefunction can be derived. This is not a homework problem it is just a thought experiment. Thank you all for your help.

I also do not understand why QFT is being brought into this. One does NOT need QFT to write down the Hamiltonian or Schrodinger equation for 2 electrons in a box. Isn't this almost a standard undergraduate QM exercise? Why the insistence of using QFT for something that we do at that level?

Zz.

 

[ftr]

I also do not understand why QFT is being brought into this. One does NOT need QFT to write down the Hamiltonian or Schrodinger equation for 2 electrons in a box. Isn't this almost a standard undergraduate QM exercise? Why the insistence of using QFT for something that we do at that level?

Zz.

I am not asking about two non-interacting electrons in a box. I am asking about two boxes at a distance from each other having and electron that includes their charges, hence their interaction.

 

[HAYAO]

I am not asking about two non-interacting electrons in a box. I am asking about two boxes at a distance from each other having and electron that includes their charges, hence their interaction.

I think he knows, I know, and everyone know.

What is the potential energy of two interacting electrons? If you know the answer to this, then you can use it as the potential energy term of the Hamiltonian to derive the wavefunctions of two interacting electrons in two separate potential boxes. Very simple.

 

[ftr]

You don't need QFT for this, and this is not a thought experiment neither.

You have two potential boxes. You have two electrons. Potential energy must be accounted for both. How would you express the potential energy of two electrons interacting? (This is almost like an answer, in fact you said yourself. You are very close to achieving Hamlitonian for the problem you are looking at!)

I think he knows, I know, and everyone know.

What is the potential energy of two interacting electrons? If you know the answer to this, then you can use it as the potential energy term of the Hamiltonian to derive the wavefunctions of two interacting electrons in two separate potential boxes. Very simple.

I think you are referring to the 1/r potential just like the hydrogen problem. But here the electron exists in a box and spatially extended , that is the problem that i see at least.

 

[HAYAO]

I think you are referring to the 1/r potential just like the hydrogen problem. But here the electron exists in a box and spatially extended , that is the problem that i see at least.

Well the 1/r potential term of hydrogen atom refers to the potential energy of the nucleus (proton) and the electron (attractive). But you can certainly extend that to the electron-electron, albeit very minor modification. This is Coulomb's law.EDIT: You understand that "r" of the potential term refers to the distance, right?

 

[ftr]

Well the 1/r potential term of hydrogen atom refers to the nucleus (proton) and the electron. But you can certainly extend that to electron-electron. This is Coulomb's law.

Yes, sure. in the hydrogen case the wave is allowed to extend to all space, however in this case the electrons are constraint to the box.

 

[HAYAO]

Yes, sure. in the hydrogen case the wave is allowed to extend to all space, however in this case the electrons are constraint to the box.

It doesn't make any difference. You are just adding another potential energy terms, i.e. the box.

 

[ftr]

It doesn't make any difference. You are just adding another potential energy terms, i.e. the box.

Well thanks. Had I known how to do it I would not have asked. Some more hints, or better yet an answer would be appreciated, a lot.

 

[ftr]

EDIT: You understand that "r" of the potential term refers to the distance, right?

yes.

 

[HAYAO]

Good, then how about attempting writing the Hamiltonian for the system? That is, the kinetic energy terms, potential energy terms (i.e. the two boxes, and the electron-electron repulsion)? I put the "s" in bold, because that is also another big hint.BTW, just so you know, the construction of the Hamiltonian of the system might look rather simple, but solving for the eigenvalues and eigenfunctions are not always simple. In principle, there are exact solutions, but mathematically it might not be very easy to get analytical solutions. This is especially true when we add potential energy terms that involve interaction between electrons (many-electron system), so some approximation may be needed.

 

[ftr]

Thanks again. this might take some time, you sound like my TA of eons ago
I think maybe I should review the electrons in solid state or something like that.

 

[hilbert2]

If the electrons move in 1 dimension and their coordinates are $x_1$ and $x_2$, the Schroedinger equation is

$-\frac{\hbar^2}{2m_e}\left(\frac{\partial^2}{\partial x_{1}^2} + \frac{\partial^2}{\partial x_{2}^2}\right)\psi (x_1 ,x_2 ) + \frac{e^2}{4\pi\epsilon_0 |x_1 - x_2 |}\psi (x_1 ,x_2) = E\psi (x_1 ,x_2 )$.

After solving this to get an infinite number of solutions, you discard those of them that are not antisymmetric in the interchange of $x_1$ and $x_2$ (you also have to include discrete spin variables to be able to do this properly), and those that don't have the property of $\psi (x_1 ,x_2 )$ being zero if $x_1$ is at a boundary point of box 1 or $x_2$ is at a boundary point of box 2 (this is how the impenetrable walls of the boxes are introduced as a boundary condition).

 

[ftr]

Thanks both Zapper and Hilbert2. I even found this which you can run interactively with many other simulations.
http://demonstrations.wolfram.com/TwoElectronsInABoxWavefunctions/

However, I am still not sure how to interpret the solutions since each electron is confined to its box and the standard solution allows exchange.Moreover, I have to check for continuity on the borders which look tough.

 

[hilbert2]

If the two boxes are far enough from each other, you can take the ground state wavefunction of non-interacting particles in separate boxes and then add the interaction as a 1st or 2nd order perturbation.

 

[HAYAO]

If the two boxes are far enough from each other, you can take the ground state wavefunction of non-interacting particles in separate boxes and then add the interaction as a 1st or 2nd order perturbation.

I second this. I was also suspecting that it will be quite difficult to obtain analytical solution for this system. Advantage of perturbation is that you can easily see how the interaction between electrons cause deviation from when there is no interaction, allowing more intuitively understandable result. I highly suggest ftr to do this since he says he have problem interpreting the solution.

 

[PeterDonis]

each electron is confined to its box

That would be true if the electrons were classical particles, but they're not.

 

[hilbert2]

I second this. I was also suspecting that it will be quite difficult to obtain analytical solution for this system. Advantage of perturbation is that you can easily see how the interaction between electrons cause deviation from when there is no interaction, allowing more intuitively understandable result. I highly suggest ftr to do this since he says he have problem interpreting the solution.

This could also be solved numerically. If the centers of the boxes are at, say, points $c_1$ and $c_2$ on the $x$-axis, and the length of both boxes is $L$, the potential $V(x_1 ,x_2 )$ could be approximated with a function that jumps to a very large but finite value when one of the electrons crosses the wall of a box and where the Coulomb term is proportional to $\frac{1}{|x_1 - x_2 |+\epsilon}$ with $\epsilon$ a very small positive number that prevents the potential from being infinite in any situation. Then we could choose an antisymmetrized initial state like

$\Psi (x_1 ,x_2 ,t_0 ) = C\left[\exp\left(-10\left(\frac{x_1 - c_1}{L}\right)^2\right)\exp\left(-10\left(\frac{x_2 - c_2}{L}\right)^2\right) - \exp\left(-10\left(\frac{x_2 - c_1}{L}\right)^2\right)\exp\left(-10\left(\frac{x_1 - c_2}{L}\right)^2\right)\right]$,

and propagate it forward with an imaginary time coordinate as in these blog posts of mine:

https://physicscomputingblog.com/20...art-5-schrodinger-equation-in-imaginary-time/
https://physicscomputingblog.com/20...ution-of-pdes-part-7-2d-schrodinger-equation/ .

Then the state would decay to an approximate ground state of the potential of the original problem (however, if this is done for a long enough time interval, the electrons would tunnel between boxes and end up in an "actual" ground state where there would be possible to find two electrons in the same box with a nonzero probability).

 

[Jilang]

Thanks both Zapper and Hilbert2. I even found this which you can run interactively with many other simulations.
http://demonstrations.wolfram.com/TwoElectronsInABoxWavefunctions/

However, I am still not sure how to interpret the solutions since each electron is confined to its box and the standard solution allows exchange.Moreover, I have to check for continuity on the borders which look tough.

Isn’t that two electrons confined to the same box?

 

[ftr]

Isn’t that two electrons confined to the same box?

this was clarified in many posts . two boxes each having an electron.

 

[HAYAO]

this was clarified in many posts . two boxes each having an electron.

He's talking about the link you provided.

This could also be solved numerically. If the centers of the boxes are at, say, points $c_1$ and $c_2$ on the $x$-axis, and the length of both boxes is $L$, the potential $V(x_1 ,x_2 )$ could be approximated with a function that jumps to a very large but finite value when one of the electrons crosses the wall of a box and where the Coulomb term is proportional to $\frac{1}{|x_1 - x_2 |+\epsilon}$ with $\epsilon$ a very small positive number that prevents the potential from being infinite in any situation. Then we could choose an antisymmetrized initial state like

$\Psi (x_1 ,x_2 ,t_0 ) = C\left[\exp\left(-10\left(\frac{x_1 - c_1}{L}\right)^2\right)\exp\left(-10\left(\frac{x_2 - c_2}{L}\right)^2\right) - \exp\left(-10\left(\frac{x_2 - c_1}{L}\right)^2\right)\exp\left(-10\left(\frac{x_1 - c_2}{L}\right)^2\right)\right]$,

and propagate it forward with an imaginary time coordinate as in these blog posts of mine:

https://physicscomputingblog.com/20...art-5-schrodinger-equation-in-imaginary-time/
https://physicscomputingblog.com/20...ution-of-pdes-part-7-2d-schrodinger-equation/ .

Then the state would decay to an approximate ground state of the potential of the original problem (however, if this is done for a long enough time interval, the electrons would tunnel between boxes and end up in an "actual" ground state where there would be possible to find two electrons in the same box with a nonzero probability).

That's a very elegant way to numerically solve it!