Energy levels for a 3D cubical box

In summary, the equation for the energies of n levels of a 3D cubical box is E = h^2/8mL^2 (n^2 sub x + n^2 sub y + n^2 sub z). The quantum numbers n_x, n_y, and n_z can be found by solving the Schrodinger equation with suitable boundary conditions and substituting separable sinusoidal solutions. These quantum numbers will appear in the energy values.
  • #1
cytochrome
166
3
I'm trying to find the energies for n levels of a 3D cubical box.

Is there a general equation for this?
 
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  • #2
From _Physics_, Wolfson and Pasachof, ISBN 0-673-39836-6, p. 1055:

E = h^2/8mL^2 (n^2 sub x + n^2 sub y + n^2 sub z)

CW
 
  • #3
Thank you, now what do I use for the n_x, n_y, n_z?
 
  • #4
Hint: try substituting separable sinusoidal solutions into the Schrodinger eqn (with suitable boundary conditions).
 
  • #5
n_x , n_y , n_z are your quantum numbers, they describe your state, when you solve the Schr. Eqn. for ψ's, which are sinusoidal functions as mentioned below, n_x,y,z will appear in the energy values.
 

1. What is the concept of energy levels for a 3D cubical box?

The concept of energy levels for a 3D cubical box refers to the quantized energy states that an electron can occupy within a three-dimensional space confined by a cubic box. This concept is a simplified model often used in quantum mechanics to study the behavior of particles in a confined space.

2. How are the energy levels determined for a 3D cubical box?

The energy levels for a 3D cubical box are determined by solving the Schrödinger equation, which describes the behavior of quantum particles. The solution to this equation yields a set of discrete energy values that correspond to different energy levels within the cubic box.

3. How does the size of the cubic box affect the energy levels?

The size of the cubic box directly affects the energy levels, as smaller boxes have a higher energy spacing between levels compared to larger boxes. This is due to the confinement of particles in a smaller space, leading to a more pronounced quantum effect.

4. Can the energy levels for a 3D cubical box be visualized?

Yes, the energy levels for a 3D cubical box can be visualized through a graphical representation known as an energy level diagram. This diagram shows the different energy levels and the corresponding electron occupancy within the box, providing a visual understanding of the quantized energy states.

5. What are the applications of studying energy levels for a 3D cubical box?

The study of energy levels for a 3D cubical box has various applications in fields such as materials science, nanotechnology, and quantum computing. It allows for the prediction and understanding of the behavior of particles in confined spaces, leading to advancements in these industries.

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