# Quantum Mechanics of a Bouncing Ball

Quantum Mechanics of a Bouncing Ball

The Schrödinger equation can be written , where is the mass of the ball (idealized as a point mass), is the acceleration of gravity, and is the vertical height (with ground level taken as ). For perfectly elastic collisions, the potential energy at can be assumed infinite: , leading to the boundary condition . Also, we should have as .

-ψ''(z)+mgzψ(z)=Eψ(z)

ℏ

2

2m

m

g

z

z=0

z=0

V(0)=∞

ψ(0)=0

ψ(z)0

z∞

The problem, as stated, is not physically realistic on a quantum level, given Earth's value of , because would have to be much too small. But an analogous experiment with a charge in an electric field is possibly more accessible. We will continue to refer to the gravitational parameters, however.

g

m

Redefining the independent variable as , the equation reduces to the simpler form . (The form of the variable is suggested by running DSolve on the original equation). The solution that remains finite as is found to be . (A second solution, , diverges as .)

x=z-

2mg

2

ℏ

2

1/3

E

mg

ψ''(x)-xψ(x)=0

x∞

ψ(x)=constAi(x)

Bi(x)

x∞

The eigenvalues can be found from the zeros of the Airy function: , using N[AiryAiZero[n]]. The roots lie on the negative real axis, the first few being approximately , , , , , , …. Defining the constant , the lowest eigenvalues are thus given by , , , and so on. The corresponding (unnormalized) eigenfunctions are . These are plotted on the graphic.

E

n

Ai-=0

2mg

2

ℏ

2

1/3

E

mg

-2.33811

-4.08795

-5.52056

-6.78671

-7.94413

-9.02265

α=

2mg

2

ℏ

2

1/3

E/mg=

0

2.33811α

-1

E/mg=4.08795α

1

-1

E/mg=5.52056α

2

-1

ψ(z)=Ai[α(z-E/mg)]

n

n