The Energy of the Trapped Electron in a One-Dimensional Space

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In summary, to calculate the length of the space L in nm where an electron is trapped, use the energy expression for a particle in a box, which includes a pi-squared term in the numerator. The given 523 nm value represents the wavelength of the absorbed photon, which can be used to calculate the energy of the photon using the formula E_{photon}=\frac{hc}{\lambda}.
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Homework Statement



Calculate the length of the space L in nm if an electron is trapped in one dimensional space of length L, and shows an absorption at 523nm due to transition from ψ2 to ψ3.


Homework Equations



Energy expression for particle in box:

Ev= (n2h2)/(8mL2) n=1, 2, 3...

The Attempt at a Solution



I don't understand what I am suppose to do with 523 nm. Once I know that I can continue. Please help with this. Thanks.
 
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First, the general formula for the energy of your eigenstates is incorrect. There should be a pi-squared term in the numerator.

Electrons can only be located in specific orbits, right? So the given 523 nm value is the wavelength of the absorbed photon. What is the energy of the photon?

[tex] E_{photon}=\frac{hc}{\lambda} [/tex]

EDIT: I changed h-bar to h.
 
Last edited:

FAQ: The Energy of the Trapped Electron in a One-Dimensional Space

1. What is the "particle in a box" model?

The "particle in a box" model is a simplified representation of the behavior of a particle in a confined space, such as an electron in an atom or a molecule in a container. It assumes that the particle is confined within a one-dimensional box and that its energy is quantized, meaning it can only have certain discrete energy levels.

2. How does the energy level of a particle in a box relate to its confinement?

The energy level of a particle in a box is directly related to the size of the box and the particle's mass. The smaller the box, the higher the energy levels and vice versa. This is because the particle's wave function is forced to fit within the boundaries of the box, resulting in quantized energy levels.

3. What is the significance of the "particle in a box" model in quantum mechanics?

The "particle in a box" model is a fundamental concept in quantum mechanics. It helps us understand the behavior of particles in confined spaces and how their energy levels are affected by confinement. This model is also used to explain the properties of electrons in atoms, which are crucial for understanding chemical bonding and the structure of matter.

4. Can the "particle in a box" model be applied to real-world systems?

Yes, the "particle in a box" model can be applied to real-world systems, such as molecules in a container or electrons in a solid material. However, it is a simplified model and does not account for other factors that may affect the behavior of particles, such as interactions with other particles or external forces.

5. How does the "particle in a box" model relate to Heisenberg's uncertainty principle?

The "particle in a box" model is related to Heisenberg's uncertainty principle because it demonstrates the limitations of knowing both the position and momentum of a particle. In this model, the particle's position is well-defined within the box, but its momentum is uncertain. This is a fundamental principle in quantum mechanics that states that the more precisely we know a particle's position, the less precisely we can know its momentum, and vice versa.

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