Electron in 1-D box: photon absorbed?

In summary, the conversation is about a problem involving the transition of an electron in a one-dimensional box to the second excited state. The ground-state energy is given and the wavelength of the photon absorbed needs to be calculated. The solution involves calculating the length of the box and the energy of the second excited state, and then using the difference between these energy levels to find the energy of the absorbed photon. The correct answer is found to be 158.532 nm.
  • #1
Ryaners
50
2
I don't know where I'm going wrong with this problem - I was so sure I had it right but the online grader tells me otherwise :oldfrown:

Homework Statement


An electron in a one-dimensional box has ground-state energy 2.60 eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?

Homework Equations


En = n2h2 / 8mL2
hf = hc / λ
⇒ λ = hc / hf

3. The Attempt at a Solution

The ground-state energy in Joules is (2.60 eV)⋅(1.602⋅10-19 J/eV) = 4.165668⋅10-19 J

First I calculated the length of the box by rearranging the energy level equation above:
L = √(n2h2 / 8mEn)

For n=1, this gives:
L = √{(6.626⋅10-34)2 / 8(9.109⋅10-31)(4.165668⋅10-19)}
= 3.80294⋅10-10 m

Then I used this L to find the energy of the n=2 level:
En=2 = {(2)2(6.626⋅10-34)2} / {8(9.109⋅10-31)(3.80294⋅10-10)2}
= 1.66627⋅10-18 J

The difference in these energy levels is:
1.66627⋅10-18 J - 4.165668⋅10-19 J = 1.2497⋅10-18 J

I took this to be equal to the energy of the photon absorbed, i.e. equal to hf. Then:
λphoton = {(6.626⋅10-34)(2.99⋅108)} / 1.2497⋅10-18 J
= 158.532 nm

I corrected it to 3 significant figures to input the answer; I tried both 159nm and 158nm in case it was a rounding error but Computer Says No. Can anyone spot where I'm going wrong? Thanks in advance!
 
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  • #2
n=2 is the first excited state, not the second one.
 
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  • #3
blue_leaf77 said:
n=2 is the first excited state, not the second one.
Ack, of course! Thanks so much, I completely missed that. :)
 

Related to Electron in 1-D box: photon absorbed?

1. What is an electron in a 1-D box?

An electron in a 1-D box is a simplified model used in quantum mechanics to study the behavior of electrons confined to a one-dimensional space. The box represents the boundaries in which the electron can exist and is used to understand the effects of quantum confinement on the electron's properties.

2. How does a photon get absorbed by an electron in a 1-D box?

A photon can get absorbed by an electron in a 1-D box when the energy of the photon matches the energy difference between two allowed energy levels in the box. This results in the electron transitioning to a higher energy level, and the photon's energy is transferred to the electron.

3. What happens to the electron after it absorbs a photon in a 1-D box?

After absorbing a photon, the electron will transition to a higher energy level. It will then remain in this new energy state until it either emits a photon to transition back to a lower energy level or is disturbed by an external force.

4. How does the energy of the absorbed photon affect the electron's behavior in a 1-D box?

The energy of the absorbed photon determines the energy level that the electron will transition to. A higher energy photon will result in the electron being in a higher energy state, and a lower energy photon will result in a lower energy state. This can affect the electron's behavior, such as its position and momentum.

5. What is the significance of studying an electron in a 1-D box absorbing a photon?

Studying an electron in a 1-D box absorbing a photon allows us to understand the principles of quantum mechanics and how particles behave in confined spaces. It also has practical applications in fields such as nanotechnology and the development of quantum devices. Additionally, it helps us understand the behavior of light-matter interactions and how energy is transferred between particles.

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