Completing the Square Homework Statement

[denjay]

Homework Statement

Consider a charged particle of mass m in a harmonic potential and in the presence also of an
external electric field E = E\hat{i}. The potential for this problem is simply

V(x) = 1/2 mw^{2}x^{2} - qεx

where q is the charge of the particle.

1) Show that a simple change of variables turns this problem into one of a particle under
only a harmonic oscillator potential. (Hint: Complete the square.)

Homework Equations

(ax-b)^{2} = a^{2}x^{2} - 2abx + b^{2}

The Attempt at a Solution

So I know the way to simplify the potential is by completing the square. I only know the way of completing the square when a quadratic equation is equal to 0 but in this case it's a function. So with that formula for (ax-b)^{2} I believe 1/2 mw^{2} is a^{2} and -2ab is -qε but I'm unsure.

So what I got was that b = qε/(2mw^{2}) so the equation is

V(x) = 1/2 mw^{2}x^{2} - qεx + q^{2}ε^{2}/2mw^{2} - q^{2}ε^{2}/2mw^{2} = (\sqrt{1/2 mw^{2}}x - qε/\sqrt{2mw^2})^2 - q^{2}ε^{2}/2mw^{2}

 

[LCKurtz]

Homework Statement

Consider a charged particle of mass m in a harmonic potential and in the presence also of an
external electric field E = E\hat{i}. The potential for this problem is simply

V(x) = 1/2 mw^{2}x^{2} - qεx

where q is the charge of the particle.

1) Show that a simple change of variables turns this problem into one of a particle under
only a harmonic oscillator potential. (Hint: Complete the square.)

Homework Equations

(ax-b)^{2} = a^{2}x^{2} - 2abx + b^{2}

The Attempt at a Solution

So I know the way to simplify the potential is by completing the square. I only know the way of completing the square when a quadratic equation is equal to 0 but in this case it's a function. So with that formula for (ax-b)^{2} I believe 1/2 mw^{2} is a^{2} and -2ab is -qε but I'm unsure.

You don't have to have an equation. Here's an example, complete the square on: $2x^2-12x$. You factor out the $2$ getting $2(x^2-6x)$. Now inside the parentheses you need a $9$ so add it and subtract it: $2(x^2-6x + 9 - 9)$ which is the same as $2(x-3)^2 - 18$. Try something like that.

 

[denjay]

Cool yeah, I did it the way you suggested and got the same answer as I did in my original post. Thanks!