What Is the Discriminant in the Schwarz Inequality Proof?

[Old Guy]

Homework Statement

Homework Equations

(\langle \alpha | + \lambda^\ast\langle \beta |)(|\alpha\rangle+ \lambda|\beta\rangle) = \langle \alpha |\alpha \rangle + |\lambda|^2\langle \beta | \beta \rangle + \lambda \langle \alpha | \beta \rangle + \lambda^\ast \langle \beta | \alpha \rangle \geq 0<br /> <br /> <br /> <br /> <h2>The Attempt at a Solution</h2><br /> This equation is the setup, and it leads to an equation that I can see is quadratic in lambda. From this, I calculate the discriminant, which must be greater than or equal to zero because all the terms are real and positive. However, when I manipulate this to get to the Schwarz inequality, I get a "less than or equal to" where I should have a "greater than or equal to". Can someone please help? Thanks.<br />

 

[Dick]

Homework Statement

Homework Equations

(\langle \alpha | + \lambda^\ast\langle \beta |)(|\alpha\rangle+ \lambda|\beta\rangle) = \langle \alpha |\alpha \rangle + |\lambda|^2\langle \beta | \beta \rangle + \lambda \langle \alpha | \beta \rangle + \lambda^\ast \langle \beta | \alpha \rangle \geq 0

The Attempt at a Solution

This equation is the setup, and it leads to an equation that I can see is quadratic in lambda. From this, I calculate the discriminant, which must be greater than or equal to zero because all the terms are real and positive. However, when I manipulate this to get to the Schwarz inequality, I get a "less than or equal to" where I should have a "greater than or equal to". Can someone please help? Thanks.

Homework Statement

Homework Equations

The Attempt at a Solution

Use tex and /tex for the tex delimiters instead of latex and latex. And you don't want to solve a quadratic. Just put in the special value lambda=-<beta|alpha>/<beta|beta>.

 

[Old Guy]

Thanks, and sorry about the equation format; I'm still trying to figure out the MathType translator.

Anyway, I understand how it works for the value of lambda you gave, but shouldn't it work for ANY value of lambda?

 

[Dick]

Thanks, and sorry about the equation format; I'm still trying to figure out the MathType translator.

Anyway, I understand how it works for the value of lambda you gave, but shouldn't it work for ANY value of lambda?

It works for any lambda, but it doesn't tell you anything very interesting for every lambda. E.g. lambda=0 isn't interesting at all.