Solving transcendental equation- Quantum Mechanics

In summary, the conversation is about solving a transcendental equation numerically using the Newton-Raphson method to find the ground state energy of a wave function in a semi-infinite well. The equation being used is tan(z)=sqrt((z(o)/z)^2-1), where z(o)=(a/h)*sqrt(2mV(o)) and z=(a/h)*sqrt(2m(E+V(o))). The value of V(o) is -10 eV and the equation is written in Maple code. The speaker is unsure if they are using the correct equation and is seeking hints on the programming procedure. They also mention the possibility of using C++ instead of Maple to write the program.
  • #1
GTdan
39
0
I don't know if I am being dumb or not but I need to solve a transcendental equation numerically and I need to write a program that can do this. The equation is so I can find the the ground state energy of a wave function in a semi-infinite well. I was told to use the Newton-Raphson method to do it and I am thinking of a transcendental equation found in the book would be the one I need to solve (if anyone thinks differently let me know). Here is the equation:

tan(z)=sqrt((z(o)/z)^2-1)

where z(o)=(a/h)*sqrt(2mV(o))

z=(a/h)*sqrt(2m(E+V(o)))

and V(o)= -10 eV

This is written in maple code btw. I'm thinking I can solve for z by using Newton's method and then solve for E afterwards since z is a function of E. That's the only idea I have that I would need Newton's method to do so can someone let me know if I am on the right track or give me some hints or something.
 
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  • #2
Okay, so you're basically after a programming procedure, since you've already decided upon using Newton-Raphson?
 
  • #3
Well, we have to use Newton Raphson. I would much rather solve it graphically like the book but that's not the case. I may have been vague but my problem is that I am I have no idea if I am going the right direction or not. Am I using the right equation to solve for E? If I am can I get some hints on the programming procedure?

Also, unless I can make Maple compile a program or something (if it even does that), I have to write it in C++.
 

1. What is a transcendental equation in quantum mechanics?

A transcendental equation in quantum mechanics is an equation that involves transcendental functions such as logarithms, exponentials, and trigonometric functions. These equations are commonly used to describe the behavior of quantum systems, such as the Schrödinger equation.

2. Why is it important to solve transcendental equations in quantum mechanics?

Solving transcendental equations in quantum mechanics is important because it allows us to understand and predict the behavior of quantum systems. These equations can help us determine the energy levels of particles, the probability of finding a particle in a certain state, and other important properties of quantum systems.

3. How do scientists solve transcendental equations in quantum mechanics?

There are a few different methods for solving transcendental equations in quantum mechanics. One common approach is to use numerical methods, such as the Newton-Raphson method, to approximate the solutions. Another method is to use analytic techniques, such as perturbation theory, to find approximate solutions.

4. Can transcendental equations have multiple solutions in quantum mechanics?

Yes, transcendental equations can have multiple solutions in quantum mechanics. This is because quantum systems can exhibit multiple energy levels and states, resulting in multiple solutions to the transcendental equations that describe them.

5. Are there any challenges in solving transcendental equations in quantum mechanics?

Yes, there are some challenges in solving transcendental equations in quantum mechanics. One challenge is that the equations can be complex and difficult to solve analytically, requiring the use of numerical methods. Additionally, the solutions to these equations may not always have a physical interpretation, making it important to carefully interpret the results.

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