{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

LinearPotential

# LinearPotential - Linear potential February 8 2008 Linear...

This preview shows pages 1–2. Sign up to view the full content.

Linear potential February 8, 2008 Linear Potential - Gravity The Schrodinger equation for the wave function of a bouncing ball is - ¯ h 2 2 m d 2 ψ dx 2 + mgxψ = (1) where we assume a perfectly elastic collision of the ball with the floor. So V ( x ) = for x < 0, and V ( x ) = mgx for V > 0. There is zero probability to find the ball at x < 0 so ψ (0) = 0. If we define a characteristic length, l 0 = ¯ h 2 2 m 2 g ! 1 3 and energy E 0 = ¯ h 2 mg 2 2 ! 1 3 then x = yl 0 and E = E 0 where y and are dimensionless and substitution into Equation 1 gives - d 2 ψ dy 2 + = ψ (2) Finally with the substitution of z = y - , Equation 2 becomes - d 2 ψ dz 2 + = 0 (3) Equation 3 is Airy’s equation and the solutions to it are Airy functions. (Some of the properties of Airy functions are summarized in Chapter 8 of Griffiths. For more details see http://dlmf.nist.gov/contents/AI/index.html) The general solution to Airy’s equation is ψ ( z ) = aA i ( z ) + bB

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

LinearPotential - Linear potential February 8 2008 Linear...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online