Basic Infinite Potential Well, Tunnelling, etc.

[blaksheep423]

I have an exam later today, and I would really apreciate it if someone could check my work or at least point out flaws in my process here. Thanks!

Homework Statement

At x=0, a proton with a kinetic energy of 10 eV is traveling in the x direction (potential energy = 0). At x=1nm, it encounters a potential barrier of height 12eV and width 0.2 nm. The potential returns to 0 at 1.2nm.

Calculate the transmission and reflection probabilities

Homework Equations

T = [1 + [V²sinh²(kL)] / [4E (V - E)]

R = 1 - T

where V = 12, E = 10, L = 2*10^-10,
and k = $$\sqrt{}2m (V-E)$$/h_bar

The Attempt at a Solution

well its all straightforward, and I got k = 7.24 * 10⁹,
which means kL is 1.45.

when I plug everything in, I get T = .12 and R = .88

I've checked it 3 times and it comes out the same every time, but this seems like a very high transmission probability, so I don't know if i made a mistake somewhere.

Homework Statement

An electron is in a 1D potential well approximated by V = 0 for 0nm < x < 2nm and V = infinity for all other x

what is the electron wave function for this lowest energy state?

what is the energy of the electron in this state

Homework Equations

shroedinger wave equation was used to derive this wave function

The Attempt at a Solution

well I got $$\Psi$$ = Asin(kx), where kL = n$$\pi$$, so
$$\Psi$$ = Asin(n$$\pi$$x / L)

when i normalize this, i get A = $$\sqrt{}2/L$$, so

$$\Psi$$ = $$\sqrt{}2/L$$sin(n$$\pi$$x / L)

so I plug in L and i end up with a numerical equation of
$$\Psi$$ = 31623sin (5$$\pi$$ * 10⁸ x)

this doesn't seem right to me.

for the energy i just plug the values into the equation:

E = h²n²$$\pi$$² / 2mL²

and i get .094 eV, which also seems wrong.

 

[Jhero]

Hello,

I am a scientist and I would be happy to review your work and provide feedback.

Firstly, your solution for the transmission and reflection probabilities appears to be correct. The high transmission probability may seem surprising, but it is actually expected for a potential barrier of this size and energy range. In general, the transmission probability increases as the energy of the particle increases and as the barrier height decreases. Your result falls within the expected range for these parameters.

For the second problem, your wave function appears to be correct. However, for the energy calculation, you should use the formula E = (n^2 * h^2)/(8mL^2), where n is the quantum number and L is the width of the potential well. This gives an energy of 0.094 eV, which is the correct answer.

Overall, your solutions seem to be correct. However, I would recommend double checking your calculations and making sure you are using the correct equations for each problem. Good luck on your exam!