Question

An idealized ping pong ball of mass m is bouncing in its ground state on a recoilless table in a one-dimensional world with only a vertical direction.

(a) Prove that the energy depends on m, g, h according to: \(\varepsilon\) = Kmg (m²g/h²)^α and determine α.

(b) By a variational calculation estimate the constant K and evaluate \(\varepsilon\) for m = 1 gram in ergs.

Answer 1

(a) By the method of dimensional analysis, if we have

then α = - 1/3. Thus, provided α = - 1/3, the expression gives the energy of the ball.

(b) Take the x coordinate in the vertical up direction with origin at the table. The Hamiltonian is

taking the table surface as the reference point of gravitational potential. Try a ground state wave function of the form \(\psi = x \exp(-\lambda x^2/2)\), where \(\lambda\) is to be determined. Consider

To minimize (H) , take \(\frac{d(H)}{d\lambda}= 0\) and obtain \(\lambda =\left(\frac{4m^2g}{3\sqrt \pi h^2}\right)^{2/3}\)

The ground state energy is then