Energy levels in finite 1d well

In summary, when solving for energy eigenvalues in a 1D finite potential well, one ends up with a transcendental equation that cannot be solved analytically. This means that numerical or graphical methods must be used to find the energy levels, instead of directly obtaining them from websites or programs. The equation involves two variables, z and z_o, which depend on the energy and potential of the particle trapped in the well. This information is referenced from Griffith's "Introduction to Quantum Mechanics" textbook.
  • #1
stone
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could anyone suggest the methods for solving the energy eigenvalues in a 1d finite potential well. are there any websites where we can directly get these instead of writing programs for rootfinding
 
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  • #2
In the process of finding the energy levels for the finite square well, you end up with a transcendental equation of the form:

[tex]\tan z=\sqrt{\frac{z_o}{z}-1}[/tex]

where:

[tex]z=\frac{a}{\hbar}\sqrt{2m(E+V_o)[/tex]

[tex]z_o=\frac{a}{\hbar}\sqrt{2mV_o}[/tex]

a= 1/2 the width of the well.

Here, z depends on the energy of the particle trapped, and z_o depends on the potential. It is impossible to solve this equation for z, so we can't analytically find E. So, we're stuck with numerical and graphical approaches in this case.

Reference for the info in this post:

Griffith's "Introduction to Quantum Mechanics"
 
  • #3


There are several methods for solving the energy eigenvalues in a 1D finite potential well. One of the most commonly used methods is the shooting method, where the wave function is numerically integrated from both sides of the well until a matching condition is met. Another approach is the matrix diagonalization method, where the Hamiltonian matrix is constructed and diagonalized to obtain the energy eigenvalues.

As for websites that provide pre-calculated energy eigenvalues for various potential wells, there are a few resources available. One example is the National Institute of Standards and Technology (NIST) Atomic Spectra Database, which has a section dedicated to potential well energies for various atoms and molecules. Another resource is the Quantum Mechanics Applet Service, which allows users to input their desired potential well parameters and calculates the corresponding energy eigenvalues.

However, it is also important to note that writing programs for rootfinding can be a valuable exercise in understanding the underlying principles and mathematics behind solving for energy eigenvalues in a 1D finite potential well. So while these websites can be helpful, I would also encourage exploring the methods and writing your own programs for a deeper understanding of the topic.
 

What is a finite 1d well?

A finite 1d well is a theoretical quantum mechanical model that describes the behavior of particles within a confined space with finite boundaries in one dimension. It is commonly used to study the energy levels and properties of systems such as atoms, molecules, and solid state materials.

How are energy levels determined in a finite 1d well?

The energy levels in a finite 1d well are determined by solving the Schrödinger equation, which describes the wave-like behavior of particles in quantum mechanics. The solutions to this equation yield discrete energy values, known as energy levels, that correspond to the allowed states of the particle within the well.

What is the significance of energy levels in a finite 1d well?

The energy levels in a finite 1d well provide important information about the behavior and properties of particles within the system. They determine the allowed energy states of the particles and can also be used to calculate other properties such as the probability of finding a particle at a specific location within the well.

How do the energy levels change in a finite 1d well when the boundaries are altered?

Changing the boundaries of a finite 1d well can significantly affect the energy levels of the system. For example, increasing the well depth or width can lead to a increase in the energy levels, while narrowing the well can cause a decrease in the energy levels. This is because the boundaries of the well act as potential barriers that affect the behavior of the particles within the well.

What real-world applications use the concept of finite 1d wells?

The concept of finite 1d wells has many applications in various fields such as solid state physics, chemistry, and materials science. It is used to understand the electronic properties of materials, the behavior of atoms and molecules, and the operation of electronic devices such as diodes and transistors. It is also a fundamental concept in the development of quantum computing and nanotechnology.

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