For a particle in a three dimensional box with Lx = Ly = Lz/3 =L. What’s the energy at the E111?
Understand the Problem
The question is asking for the energy of a particle in a three-dimensional box at the quantum state denoted as E111. The dimensions of the box are given as Lx, Ly, and Lz, where Lz is three times L. To find the energy, we will apply the formula for the energy levels of a particle in a 3D box, which depends on the quantum numbers and the dimensions of the box.
Answer
The energy of the particle in the state E111 is given by: $$ E_{111} = \frac{19 h^2}{72 m L^2} $$
Answer for screen readers
The energy of the particle in the state E111 is given by:
$$ E_{111} = \frac{19 h^2}{72 m L^2} $$
Steps to Solve
- Identify the Quantum Numbers
The quantum state E111 indicates that the quantum numbers for the particle are ( n_x = 1 ), ( n_y = 1 ), and ( n_z = 1 ).
- Dimensions of the Box
We are given that the dimensions of the box are ( L_x = L ), ( L_y = L ), and ( L_z = 3L ).
- Energy Formula for a Particle in a 3D Box
The energy levels of a particle in a three-dimensional box are given by the formula:
$$ E_{n_x n_y n_z} = \frac{h^2}{8m} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2} \right) $$
- Substitute the Values into the Formula
Substituting the quantum numbers and dimensions into the energy formula:
$$ E_{111} = \frac{h^2}{8m} \left( \frac{1^2}{L^2} + \frac{1^2}{L^2} + \frac{1^2}{(3L)^2} \right) $$
- Calculate Each Term
Now we can simplify the terms inside the parentheses:
$$ E_{111} = \frac{h^2}{8m} \left( \frac{1}{L^2} + \frac{1}{L^2} + \frac{1}{9L^2} \right) $$
Combining the terms gives:
$$ E_{111} = \frac{h^2}{8m} \left( \frac{2}{L^2} + \frac{1}{9L^2} \right) = \frac{h^2}{8m} \left( \frac{18 + 1}{9L^2} \right) = \frac{h^2}{8m} \left( \frac{19}{9L^2} \right) $$
- Final Energy Expression
Thus, the energy for the state E111 becomes:
$$ E_{111} = \frac{19 h^2}{72 m L^2} $$
The energy of the particle in the state E111 is given by:
$$ E_{111} = \frac{19 h^2}{72 m L^2} $$
More Information
This equation represents the quantized energy levels of a particle confined in a three-dimensional potential well, where ( h ) is Planck's constant, and ( m ) is the mass of the particle. The dimensions of the box contribute to the energy levels, showing that in quantum mechanics, even a simple structure like a box can lead to complex energy states.
Tips
- Incorrectly substituting dimensions: It's common to forget to incorporate the correct dimensions when calculating energy.
- Miscalculating the term combinations: Ensure to carefully combine the fractions; otherwise, the final answer might be incorrect.
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