# Transcendental equation from a finite square well potential

• user3
In summary: There is a condition that you need to satisfy in order for the solutions to be valid. This condition is that the wave function must be symmetric about the center of the well.
user3
if I have a transcendental equation such as this one: tan(l a) = -l / sqrt (64/a^2 - l^2 ) Where
l=sqrt(2m(E+V) /hbar^2 ) and 'a' is the width of a finite square well, how can I solve this equation in terms of both l and a. I have successfully graphed the two sides of the equation together by assuming 'a' to be any constant, i.e 1 . but how can I graphically solve it and extract an exact solution for E that includes a.

note: there are supposed to be three bound states:3 different solutions for E

It might help to let ##z = a l## and express the transcendental equation in terms of ##z## alone.

Note: For this type of problem you generally get two different transcendental equations. One for wave functions that are symmetric about the center of the well and one for anti-symmetric functions.

letting x = la, I got this http://www.wolframalpha.com/input/?i=tan(x)+and+-x+/+sqrt(64+-+x^2)

but i only have 2 bound states .

here's the question, please tell me what I am doing wrong:

V = infinity for x<0 and V= -Vo = -32hbar^2 / ma^2 for 0<x<a and V= 0 for x>a

I let l = sqrt(2m(E+Vo)/hbar^2) and k = sqrt(-2mE/hbar^2)

from the continuity of ψ at a: Be^(-ka) = Asin(la) (1)

and from the continuity of dψ/dx at a : -kBe^(-ka) = lAcos(la) (2)

divide (1) by (2) to get -l/k = tan(la) (3)

but k^2 + l^2 = 2mVo / hbar^2 , so k = sqrt(64/a^2 - l^2)

put the new k back in (3) to get tan(la) = -l / sqrt(64/a^2 - l^2)

finally let x = la ----> tan(x) = -x / sqrt(64 -x^2)

from which i get only two intersections on the positive y axis.

"Note: For this type of problem you generally get two different transcendental equations. One for wave functions that are symmetric about the center of the well and one for anti-symmetric functions."

TSny, the potential function is not even in this problem so I think it's only one transcendental. equation

Oh, I didn't know that you were taking V = ∞ for x<0.

Your work looks correct to me. There are in fact 3 solutions. The third one is difficult to pick up on your graph. Can you find it by regraphing?

Here's a trick I've seen. By squaring both sides of your transcendental equation and using some trig identities, show that the solutions must satisfy

##|\sin (z)| = z/8##

The graph of this is easier to analyze. However, you only want the solutions where you also satisfy the condition that ##\tan (z) <0##.

"The graph of this is easier to analyze. However, you only want the solutions where you also satisfy the condition that tan(z)<0. "

but where does this requirement come from?

user3 said:
"The graph of this is easier to analyze. However, you only want the solutions where you also satisfy the condition that tan(z)<0. "

but where does this requirement come from?

Look at your original transcendental equation.

1 person

## 1. What is a finite square well potential?

A finite square well potential is a potential energy function that describes the energy of a particle inside a finite square well, which is a potential well with finite depth and width. It is commonly used in quantum mechanics to study the behavior of particles in confined spaces.

## 2. What is a transcendental equation?

A transcendental equation is an equation that contains a transcendental function, such as trigonometric, exponential, or logarithmic functions. These equations do not have a finite number of solutions and often require numerical methods to find their roots.

## 3. How is a transcendental equation related to a finite square well potential?

A transcendental equation is used to find the allowed energy levels of a particle in a finite square well potential. The equation relates the energy of the particle to the potential energy function and the characteristics of the well, such as its depth and width.

## 4. What are the solutions of a transcendental equation from a finite square well potential?

The solutions of a transcendental equation from a finite square well potential are the allowed energy levels of the particle inside the well. These solutions can be found by solving the transcendental equation using numerical methods or by graphically analyzing the equation.

## 5. How does the behavior of a particle change with different parameters in a finite square well potential?

The behavior of a particle inside a finite square well potential is affected by the parameters of the well, such as its depth and width. Changing these parameters can alter the allowed energy levels of the particle, leading to different behaviors such as bound states, reflection, or transmission.

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