Transcendental equation from a finite square well potential

[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

 

[TSny]

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.

 

[user3]

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

 

[TSny]

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 $\tan (z) <0$.

 

[user3]

"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?

 

[TSny]

"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.