Quantum - Electron in an infinite rectangular prism well

In summary, the question asks for the value of d=b/c that minimizes the first excited state of an electron in an infinite rectangular prism well. The provided equation for energy, E=(h^2*pi^2/(2m))*((n_x/l_x)^2*(n_y/l_y)^2*(n_z/l_z)^2), is not the correct one to use for this problem. The correct equation involves separation of variables and has the terms in the denominator summed rather than multiplied. The OP's attempted solution involved plugging the values into the incorrect equation, but it seemed to work out.
  • #1
golmschenk
36
0

Homework Statement


If an electron is in an infinite rectangular prism well, with sides of length a, b, and c where c is the shortest and (b^2)*c=a^3, for what value of the d=b/c is the first excited state of the electron minimized? This isn't the complete problem but it's the part that's giving me trouble/

Homework Equations


I'm using the equations E=(h^2*pi^2/(2m))*((n_x/l_x)^2*(n_y/l_y)^2*(n_z/l_z)^2) but it's suppose to be used for an electron gas in a solid. The question is referring to a crystal. Is this equation one I want to use? Sorry for not using the correct math notation to make it look nice.

The Attempt at a Solution


My attempted solution is basically just plugging it into that equation.

Thanks for your time.
 
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  • #2
golmschenk said:

Homework Statement


If an electron is in an infinite rectangular prism well, with sides of length a, b, and c where c is the shortest and (b^2)*c=a^3, for what value of the d=b/c is the first excited state of the electron minimized? This isn't the complete problem but it's the part that's giving me trouble/

Homework Equations


I'm using the equations E=(h^2*pi^2/(2m))*((n_x/l_x)^2*(n_y/l_y)^2*(n_z/l_z)^2) but it's suppose to be used for an electron gas in a solid. The question is referring to a crystal. Is this equation one I want to use? Sorry for not using the correct math notation to make it look nice.

Thanks for your time.

i'm not that familiar with the equation you give, but don't think its the the correct one to use

the infinite potential rectangular box has a reasonable analytic solution solved through separation of variables (the spatial cartesian variables can be separated in the DE )

you will get a similar equation to the one you quote for energy, however the [itex] \frac({n_x}{L_x})^2[/itex] terms are summed not multiplied
 
  • #3
Oops, yeah, summed is actually what I meant to type. And I think that ended up working out. I don't know for sure that I got the right answer yet, but it seemed to work. Thanks.
 

What is a "Quantum - Electron in an infinite rectangular prism well"?

A quantum - electron in an infinite rectangular prism well is a model that describes the behavior of an electron confined in an infinitely deep and infinitely wide potential well in the shape of a rectangular prism. It is a simplified representation of the behavior of electrons in a solid material.

What is the significance of studying the behavior of electrons in a rectangular prism well?

Studying the behavior of electrons in a rectangular prism well allows us to understand the properties and behavior of electrons in solid materials. It also helps us understand the principles of quantum mechanics and how particles behave in confined spaces.

How does the size of the rectangular prism well affect the behavior of the electron?

The size of the rectangular prism well has a direct impact on the energy levels and wave functions of the electron. A smaller well leads to more confined energy levels and a larger well leads to more spread-out energy levels. Additionally, the shape of the well can affect the symmetry of the electron's wave function.

What is the significance of the infinite depth of the potential well in this model?

The infinite depth of the potential well is an idealization that allows us to simplify the calculations and focus on the behavior of the electron within the confines of the well. In reality, potential wells in solid materials have finite depths and the behavior of the electron is more complex.

How does the behavior of an electron in a rectangular prism well differ from that of a free electron?

An electron in a rectangular prism well is confined to a specific region and therefore has discrete energy levels and a quantized wave function. A free electron, on the other hand, is not confined and can have a continuous range of energy levels and a non-quantized wave function.

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