Kinetic energy of electron & energy states

[Kognito]

Homework Statement

Use the relationship kinetic energy E = p^2/2m to show that the energy $E_{0}$ of an electron of mass m in its lowest energy state is given by $E_{0}$ = h^2/8mL^2

Homework Equations

E = p^2/2m

$E_{0}$ = h^2/8mL^2

The Attempt at a Solution

I've stared at this for far too long, googled it to death as well as checked through all my course materials and I can't seem to get started with it. Any ideas at all would be welcome.

Thank you in advance,
Kognito

 

[Pagan Harpoon]

What is L here? Is the electron in a 3d box and L is the side length? Does the problem say what boundary conditions it wants you to apply at the box edges?

 

[Kognito]

My apologies, I missed that off the end of the sentence (too busy messing with equation formatting controls).

But yes, L represents a box within which the electron resides. The question doesn't specifically mention boundary conditions though the previous part of the question said to draw the electron in its lowest energy state, able to move freely along the length of the box, as represented by a standing wave.

 

[Pagan Harpoon]

There's two things you can do... one is write down the Schrodinger equation and then solve it.

The other is to say that the electron's wavefunction has to have nodes at the box edges, so write down an expression for its deBroglie wavelength and apply the appropriate constraints to it.

 

[Nickie]

Hello Kognito,

Firstly, let's define some terms and concepts. Kinetic energy is the energy an object possesses due to its motion. In this case, we are considering the motion of an electron, which has a mass m. The momentum of an object is defined as its mass multiplied by its velocity, or p = mv. In the case of an electron, its velocity is often represented by the symbol v, but we can also express it in terms of its wavelength, λ, using the formula v = λf, where f is the frequency.

Now, let's look at the first equation given: E = p^2/2m. This is the formula for kinetic energy, where p represents momentum and m represents mass. We can rewrite this equation as E = (mv)^2/2m. Using the definition of momentum, we can substitute mv for p, giving us E = (p)^2/2m.

Next, we can look at the second equation given: E_{0} = h^2/8mL^2. This equation represents the energy of an electron in its lowest energy state, where h represents Planck's constant and L represents the length of the box in which the electron is confined.

Now, let's combine these two equations. We can equate E and E_{0} since they both represent the energy of the electron. This gives us (p)^2/2m = h^2/8mL^2.

To solve for p, we can rearrange the equation to get p = √(h^2/4mL^2).

Next, we can substitute this value for p back into our original equation for kinetic energy. This gives us E = (√(h^2/4mL^2))^2/2m. Simplifying, we get E = h^2/8mL^2, which is the same as our equation for E_{0}. This shows that the kinetic energy of an electron in its lowest energy state is equal to the energy of the electron in its lowest energy state, proving the relationship given in the homework statement.

I hope this helps! Let me know if you have any further questions. Keep up the good work as a scientist!